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62 KiB
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{"Definitions/Cauchy-Sequence":{"title":"Cauchy Sequence","links":[],"tags":[],"content":"Definition\nA Cauchy sequence is a sequence where the elements become arbitrarily close to each other as the sequence progresses.\nExamples\nCauchy Sequence\n\\Sigma_{n=1}^\\infty \\frac{1}{n^2} = 1, \\, \\frac{1}{4}, \\, \\frac{1}{9}, \\, \\dots\n\\lim_{ n \\to \\infty } \\frac{1}{n^2} = 0 \nWhich this sequence converges to 0, towards infinity\nNon-Cauchy Sequence\n\\Sigma_{n=1}^{\\infty}(-1)^n = -1, \\, 1, \\, -1, \\, 1, \\, \\dots\nThese never converge to a limit, hence it is not Cauchy.\nFurthermore, here, using something like \\lim_{ n \\to \\infty } (-1)^n is nearly impossible to know what the value would be as \\infty is neither even or odd."},"Definitions/Cauchy-Schwarz-Inequality":{"title":"Cauchy-Schwarz Inequality","links":["Definitions/Terminology/QED"],"tags":[],"content":"Definition\n| (u|v) | \\leq \\| \\, u \\, \\|_{2} \\times \\| \\, v \\, \\|_{2}\n\\| \\, u + v \\, \\|_{2} \\leq \\| \\, u \\, \\|_{2} + \\| \\, v \\, \\|_{2}\n\n\n \n Proof of Cauchy-Schwartz \n \n \n\nInsert a \\equiv - \\frac{\\overline{(u | v)}}{\\| \\, u \\, \\|_{2}^2} for u \\neq 0 into\nf(a) \\equiv |a|^2 \\times \\| \\, u \\, \\|_{2}^2 + \\text{Re}(a \\times (u | v)) + \\| \\, v \\, \\|_{2}^2 = \\| \\, au + v \\, \\|_{2}^2 \\geq 0\nQED\n\n"},"Definitions/Functions/Characteristic-Function":{"title":"Characteristic Function","links":[],"tags":[],"content":"\\varkappa_{y} \\in \\Pi_{X} \\, \\{ 0, \\, 1 \\}, so \\varkappa_{y} : X \\to \\{ 0, \\, 1 \\} defined as:\n1, & \\text{if}\\ x \\in Y\\\\ 0, & \\text{if}\\ x \\notin Y\n\\end{cases}$$"},"Definitions/Functions/Direct-Product":{"title":"Direct Product","links":[],"tags":[],"content":"X \\times Y = \\{ x: \\{ 1, 2 \\} \\to x_{1} \\cup x_{2} | x_{1} X_{1} \\wedge x_{2} \\in X_{2} \\}\nwhere X = X_{1} and Y = X_{2}, basically (X_{1}, X_{2})."},"Definitions/Functions/Inverse-Function":{"title":"Inverse Function","links":[],"tags":[],"content":"f^{-1} : Y \\to X\nsuch that\nff^{-1} = I_{x} \\; \\land f^{-1}f = I_{y}\n\\exists f^{-1} \\iff f \\; \\text{bijective}"},"Definitions/Functions/Metric":{"title":"Metric","links":["Definitions/Metric-Spaces/Metric-Space"],"tags":[],"content":"Definition\nA function which measures distances between two points in a Metric Space."},"Definitions/Functions/Power-Set":{"title":"Power Set","links":[],"tags":[],"content":"Definition\n\\wp(X) of X consists of all the subsets of X.\nAmount of Elements in a Power Set\nLets say we have |X|:\n|X| = |\\{ 1, \\, \\dots, \\, n \\}|\nThe \\wp(X) would have 2^{n} elements in the set.\nExample\nX = \\{ 1, \\, 2 \\}\n\\wp(X) = \\{ \\emptyset, \\, \\{ 1 \\}, \\, \\{ 2 \\}, \\, X \\}"},"Definitions/Hilbert-Spaces":{"title":"Hilbert Spaces","links":["Banach-Space","Definitions/Inner-Product"],"tags":[],"content":"Definition\nThese are Banach Spaces with norms given by an Inner Product.\n\n\n \n The norm is defined as: \n \n \n\n\\| \\, v \\, \\|_{2} \\equiv \\sqrt{ (v | v) }\nd(u, v) = \\| \\, u - v \\, \\|_{2}\n\n"},"Definitions/Inner-Product":{"title":"Inner Product","links":[],"tags":[],"content":"Definition\n(\\cdot | \\cdot) : V \\times V \\to \\mathbb{C}\n\n\n \n Info\n \n \n\nV \\times V \\ni (u, v) \\mapsto (u | v) \\in \\mathbb{C}\nSuch that (au + bv | w) = a(u | w) + b(v|w)\nand \\overline{(u | v)} = (v | u)\n\n\n \n (w | au + bv) = \\overline{(au + bv | w)} = \\overline{a(u | w) + b (v | w)} = \\bar{a} \\overline{(u | w)} + \\bar{b} \\overline{( v | w)} = \\dots\n \n \n\nand (v | v) \\geq 0 and (u | u) = 0 \\implies u = 0\n\n"},"Definitions/Least-Upper-Bound-Property":{"title":"Least Upper Bound Property","links":[],"tags":[],"content":"X \\subset \\mathbb{R}\n\\exists c \\in \\mathbb{R} \\; \\text{such that}\nx \\lt c, \\forall x \\in X\nsup(\\langle 0, 1 \\rangle) = 1 \\notin \\langle 0, 1 \\rangle\nsup(\\langle 0, 1 ] \\,)\nsup(\\mathbb{R}) = \\infty"},"Definitions/Linear-Map":{"title":"Linear Map","links":[],"tags":[],"content":"Definition\nL: V \\to \\mathbb{C}^n \\; \\text{bijective}\nL (ax + by) = a \\times L(x) + b \\times L(y), \\; \\forall x, \\, y \\in V, \\, a; \\, b \\in \\mathbb{C}\n\\| \\, L(x) \\, \\| = \\| \\, x \\, \\|, \\; (L(x) | L(y)) = (x | y)"},"Definitions/Measure-Theory/Sigma-Algebra/Borel-Measurable":{"title":"Borel Measurable","links":["Definitions/Topological-Spaces/Continuous"],"tags":[],"content":"Definition\nSay f : X \\to Y is continuous, \\implies f is Borel measurable."},"Definitions/Measure-Theory/Sigma-Algebra/Borel-Sets":{"title":"Borel Sets","links":["Definitions/Sets/Open-Sets","Definitions/Topological-Spaces/Topological-Space"],"tags":[],"content":"Definition\nThe \\sigma-algebra generated by the open sets in a topological space X is the \\sigma-algebra of Borel Sets of X."},"Definitions/Measure-Theory/Sigma-Algebra/Measurable":{"title":"Measurable","links":["Definitions/Topological-Spaces/Topological-Space"],"tags":[],"content":"Definition\nf: X \\to Y (Topological Space) is measurable if f^{-1}(V) \\in M \\forall \\text{ open } V \\subset Y."},"Definitions/Measure-Theory/Sigma-Algebra/Measure":{"title":"Measure","links":[],"tags":[],"content":"Definition\nA measure on X is a function \\mu : M \\to [0,\\infty] such that:\n\n\\mu(\\emptyset) = 0\n\\mu(\\cup^{\\infty}_{n=1}A_{n}) = \\Sigma^{\\infty}_{n=1} \\mu(A_{n}) (for pairwise disjoint A_{n} \\in M)\nThen we say X is a measure space.\n"},"Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra":{"title":"Sigma-Algebra","links":["Definitions/Measure-Theory/Sigma-Algebra/Measurable","Definitions/Measure-Theory/Sigma-Algebra/Measure"],"tags":[],"content":"Definition\nA \\sigma-algebra in a set X is a collection of subsets, so called measurable sets, of X such that (the requirements are):\n\nX \\in M\nA \\in M \\implies A^{\\complement} \\in M (X^{\\complement} = \\emptyset \\in M)\nA_{n} \\in M \\implies \\cup^{\\infty}_{n=1} A_{n} \\in M (\\implies \\cap^{\\infty}_{n=1} A_{n} = (\\cup^{\\infty}_{n=1}A^{\\complement})^{\\complement} \\in M))\n\nRelated Terminologies/Functions\n\nMeasurable\nMeasure\n"},"Definitions/Metric-Spaces/Ball":{"title":"Ball","links":["Definitions/Functions/Metric"],"tags":[],"content":"Definition\nLet d be a metric on a set X.\nThe (open) ball with centre x \\in X and radius r \\geq 0 is B_{r} \\equiv \\{ y \\in X | d(x,y) \\gt r\\}.\nA sequence \\{ X_{n} \\} in X converges to x \\in X if it eventually belongs to any ball B_{r}(x); \\forall r \\gt 0 \\; \\exists N \\in \\mathbb{N} such that \\underbrace{d(x, x_{n})}_{x_{n} \\in B_{r}(x)} \\lt r, \\; \\forall n \\gt N."},"Definitions/Metric-Spaces/Interior-Point":{"title":"Interior Point","links":[],"tags":[],"content":"Definition\nPoints such that small enough balls centred around them are contained in A."},"Definitions/Metric-Spaces/Metric-Space":{"title":"Metric Space","links":["Definitions/Functions/Metric"],"tags":[],"content":"Definition\nA Metric on a set X is a function d : X \\times X \\to [ \\, 0, \\infty \\rangle such that\n\nd(x,y) = d(y, x), \\; \\forall x,y \\in X\nd(x, y) = 0 \\iff x = y\nd(x,z) \\leq d(x,y) + d(y, z) (think of this as a triangle and Pythagoras’ Theorem)\nThink of d(x, y) as the distance between x and y.\n"},"Definitions/Nets":{"title":"Nets","links":[],"tags":[],"content":"\\{ x_{i} \\}_{i \\in I} \\; \\text{NET} \\; \\underbrace{I}_{\\text{VFOs}} \\to X\nx \\in \\overline{X} \\iff \\exists \\; \\text{NET} \\; \\underbrace{x_{i}}_{\\in X} \\to x\nI = neighbourhoods of x with A \\geq B \\iff A \\subset B.\n\\{ \\emptyset, X \\} - all nets in X will converge to all points in X."},"Definitions/Norm":{"title":"Norm","links":[],"tags":[],"content":"Definition\nA norm on V is a map \\|\\cdot\\| : V \\to [ \\, 0, \\infty \\rangle such that\n\n\\| u + v \\| \\leq \\| u \\| + \\| v \\|, \\; \\forall u,v \\in V\n\\| c \\times v \\| = | c | \\times \\| v \\|, \\; \\forall v \\in V, \\, \\forall c \\in \\mathbb{C}\n\\| v \\| = 0 \\implies v = 0\n\nThink of \\| v \\| as the length of v.\nNorm of 0\n\\| 0 \\| = \\| 0 \\times u \\| = \\| c \\times u \\| = \\| 0 \\| \\times \\|u \\| = 0 \\times \\| u \\| = 0"},"Definitions/Number-Field":{"title":"Number Field","links":[],"tags":[],"content":"a \\neq 0, \\; \\frac{1}{a}\nab = ba\na(b+c) = ab + ac\na \\gt b \\iff a+1 \\gt b + 1"},"Definitions/Period-of-a-Fraction":{"title":"Period of a Fraction","links":[],"tags":[],"content":"Definition\nThe period of a fraction is the digits that repeat themselves.\nThe digits can be from 0-9, and the length of the period is determined by the denominator, n in \\frac{a}{n} .\nExamples\n\\frac{22}{7}= 3.142857142857\\dots where the period here is 142857 and has a length 6\n\\frac{1}{2} = 0.5000000\\dots has the period 0, with the length being 1."},"Definitions/Rational-Cauchy-Sequences":{"title":"Rational Cauchy Sequences","links":[],"tags":[],"content":"Definition\nThey are sequences \\{ x_{n} \\subset \\mathbb{Q} \\} such that\n\\forall k \\in \\mathbb{N} \\; \\exists N \\in \\mathbb{N}\nSuch that\n|x_{m} - x_{n}| \\lt \\frac{1}{k}, \\; \\forall m, n \\gt N."},"Definitions/Sets/Complex-Numbers":{"title":"Complex Numbers","links":[],"tags":[],"content":"z = a + ib\nz = r(\\cos \\theta + i\\sin \\theta)\nz = re^{i \\theta}\nOperations\nMultiplying Vectors\nz_{1} \\times z_{2} = (r_{1} \\times r_{2})(\\cos(\\theta_{1} + \\theta 2) + i \\sin (\\theta_{1} + \\theta_{2}))"},"Definitions/Sets/Open-Cover":{"title":"Open Cover","links":["Definitions/Sets/Open-Sets"],"tags":[],"content":"Definition\nOpen Sets with union X"},"Definitions/Sets/Open-Map":{"title":"Open Map","links":["Definitions/Sets/Open-Sets"],"tags":[],"content":"Definition\nAn open map takes one open set and maps it to another open set."},"Definitions/Sets/Open-Sets":{"title":"Open Sets","links":["Definitions/Metric-Spaces/Interior-Point","Definitions/Sets/Open-Sets"],"tags":[],"content":"Definition\nAn open set A \\subset X in (X, d) (means set with a metric) consists only of interior points.\nThen a sequence converges to x \\in X \\iff it eventually belongs to any open set containing x."},"Definitions/Statements/And":{"title":"And","links":[],"tags":[],"content":"P \\land Q\nTruth Table\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nPQ=FFFFTFTFFTTT"},"Definitions/Statements/Implies":{"title":"Implies","links":[],"tags":[],"content":"P \\implies Q\nTruth Table\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nPQ=FFTFTTTFFTTT"},"Definitions/Statements/Not":{"title":"Not","links":[],"tags":[],"content":"¬P\nThe symbol (¬) is called negation\nTruth Table\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nP=FTTF"},"Definitions/Statements/Or":{"title":"Or","links":[],"tags":[],"content":"P \\lor Q\nTruth Table\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nPQ=FFFFTTTFTTTT"},"Definitions/Subcover":{"title":"Subcover","links":[],"tags":[],"content":"Definition\nAny subcollection with union X"},"Definitions/Subnet":{"title":"Subnet","links":["Definitions/Nets","Drawing-2025-02-13-11.47.33.excalidraw"],"tags":[],"content":"Definition\nA subnet of a net f: I \\to X is a net g: J \\to X and a map h : J \\to I such that g = f \\circ h and such that \\forall i \\in I \\exists j \\in J with h(j') \\geq i \\; \\forall j' \\geq j.\nExample\n\n\n \n Example\n \n \n\nTransclude of Drawing-2025-02-13-11.47.33.excalidraw\nSequence:\nf : I = \\mathbb{N} \\to \\mathbb{R} : f(n) = x_{n}\nSubsequence:\n\\{ \\underbrace{x_{1}}_{= x_{1}}, \\, \\underbrace{x_{3}}_{= x_{2}}, \\, \\underbrace{x_{5}}_{= x_{3}}, \\, \\underbrace{x_{7}}_{= x_{4}}, \\, \\dots \\} \\to 1\nDefinition:\ng : J = \\mathbb{N} \\to \\mathbb{R} by g(j) = x'_{j} = x_{2j-1} = f(2j-1) = f \\circ h(j), where h : J \\to I is defined by h(j) = 2j-1.\n\n"},"Definitions/Terminology/Algebraically-Complete":{"title":"Algebraically Complete","links":[],"tags":[],"content":"For Complex Numbers\nAny equation\na_{n} z^n + \\dots + a_{1} z^1 + a_{0} = 0\nwith a_{i} \\in \\mathbb{C} has n roots, or solutions, counted with multiplicity."},"Definitions/Terminology/Bijective":{"title":"Bijective","links":["Definitions/Terminology/Injective","Definitions/Terminology/Surjective"],"tags":[],"content":"Definition\nIt is both Injective and Surjective.\nBijective Map\nAn example could be\nX = \\{ 0,\\, L \\} \\simeq \\{ \\text{Person}, \\, \\text{House} \\}\nWhere 0 would map to \\text{Person} and L would map to \\text{House}.\nAnother example could be\n\\{ 1, \\dots, 5 \\} \\xrightarrow{f^{\\text{\\; Bijective}}} X"},"Definitions/Terminology/Bounded":{"title":"Bounded","links":[],"tags":[],"content":"Definition\n\\exists M such that \\| \\, x \\, \\| M, \\; \\forall x \\in A.\n"},"Definitions/Terminology/Countable":{"title":"Countable","links":[],"tags":[],"content":"What is countable?\nNatural Numbers\nQuite simple\n\\mathbb{N} = \\{ 1, 2, 3, 4, \\dots \\}\nWhich you can just add it up every time\nInteger Numbers\nZ can be countable as you can go\n1 \\to 0\n2 \\to 1\n3 \\to -1\n4 \\to 2\n5 \\to -2\nand so on…\nHere they start at 0, then go 1, -1, 2, -2, etc…\nRational Numbers\nThey are countable, but it is a lot more work to show"},"Definitions/Terminology/Injective":{"title":"Injective","links":[],"tags":[],"content":"f: \\mathbb{R} \\to \\mathbb{R}\nUsually a function would be mapped like:\nG(f) = \\{ (x, f(x)) | x \\in X \\}\nDefinition\nA horizontal line going through a function (on a graph) should only intersect the function only once"},"Definitions/Terminology/QED":{"title":"QED","links":[],"tags":[],"content":"Can be written as “Q.E.D.” or “QED”.\nIt is shortened in Latin from “quod erat demonstrandum” (that which was to be demonstrated).\nDefinition\nA notation which is often placed at the end of a mathematical proof to indicate its completion"},"Definitions/Terminology/Surjective":{"title":"Surjective","links":[],"tags":[],"content":"y \\in Y\n\\exists x \\in X\n\\text{such that} \\; f(x) = y"},"Definitions/Topological-Spaces/Continuous":{"title":"Continuous","links":["Definitions/Sets/Open-Sets","Definitions/Terminology/Bijective","Homeomorphic","Definitions/Topological-Spaces/Topological-Space"],"tags":[],"content":"Definition\nWe say f is continuous (at every x) if f^{-1}(A) \\equiv \\{ x \\in X | f(x) \\in A \\} is open for every open A \\subset Y.\nWe say f is open if f(B) is open and \\forall open B \\subset X.\nIf f is a bijection that is both continuous and open, it is a homeomorphism, and X and Y are homeomorphic, written X \\simeq Y; they are the ‘same’ as topological spaces.\nIn-depth Definition\nA function f : X \\to Y between topological spaces is continuous at x \\in X if for every neighbourhood A of f(x), we can find a neighbourhood B of x such that f(B) \\subset A, or B \\subset f^{-1}(A)."},"Definitions/Topological-Spaces/Hausdorff":{"title":"Hausdorff","links":["Definitions/Topological-Spaces/Topological-Space"],"tags":[],"content":"Definition\nIf there are two points x and y in a topological space X that can be separated by neighbourhoods if there exists a neighbourhood U of x and a neighbourhood V of y, such that U and V are disjoint U \\cup V = \\emptyset.\nX is a Hausdorff space if any two distinct points in X are separated by neighbourhoods."},"Definitions/Topological-Spaces/Induced/Initial-Topology":{"title":"Initial Topology","links":["Definitions/Topological-Spaces/Topological-Space","Definitions/Topological-Spaces/Induced/Weakest-Topology","Definitions/Topological-Spaces/Continuous"],"tags":[],"content":"Definition\nThe initial topology on X induced by a family of functions f : X \\to Y_{f} into topological spaces Y_{f} is the weakest topologyon X making all these functions continuous.\nHere: F = \\{ f^{-1} \\; | \\; f : X \\to Y_{f}, \\; A \\; \\text{open in} \\; Y_{f} \\}."},"Definitions/Topological-Spaces/Induced/Product-Topology":{"title":"Product Topology","links":["Definitions/Topological-Spaces/Topological-Space","Definitions/Topological-Spaces/Induced/Initial-Topology","Drawing-2025-02-24-12.47.32.excalidraw"],"tags":[],"content":"Definition\nThe product topology on \\Pi X_{\\lambda}, X_{\\lambda} topological spaces, is the initial topology induced by the family of projections \\Pi_{\\lambda}\n\n\n \n What is \\Pi_{\\lambda}?\n \n \n\n\\pi_{\\lambda} : \\Pi_{\\lambda' \\in \\wedge} X_{\\lambda'} \\to X_{\\lambda}\n\\pi_{\\lambda}(f) = f(\\lambda)\n\\pi_{\\lambda}((X_{\\lambda'})) = x_{\\lambda}\n\n\n\\underbrace{\\Pi_{\\lambda \\in \\wedge} X_{\\lambda} \\equiv}_{\\in (x_{\\lambda})_{\\lambda \\in \\wedge}} \\{ f : \\wedge \\to \\cup_{\\lambda \\in \\wedge} X_{\\lambda} \\; | \\; \\underbrace{f(\\lambda)}_{x_{\\lambda}} \\in X_{\\lambda} \\}\nExample\nProduct of 2 topological spaces: x_{1} \\times x_{2}\n= \\{ (x_{1}, x_{2}) \\; | \\; x_{i} \\in X_{i} \\}\n\\pi_{1}((x_{1}, x_{2})) = x_{1}\nTransclude of Drawing-2025-02-24-12.47.32.excalidraw"},"Definitions/Topological-Spaces/Induced/Separating-Points":{"title":"Separating Points","links":[],"tags":[],"content":"Definition\nA family F of functions on a set separating points x \\neq y in the set if f(x) \\neq f(y) for some f \\in F"},"Definitions/Topological-Spaces/Induced/Weakest-Topology":{"title":"Weakest Topology","links":["Definitions/Topological-Spaces/Topology"],"tags":[],"content":"Definition\nGiven F \\subset \\wp(X). The weakest topology on X that contains F is the intersection of all the topologies that contains F. This is a topology, and consists of \\emptyset, X, and all unions of finite intersections of members from F.\n\n\n \n Example\n \n \n\nF \\subset \\tau\n\\textvisiblespace \\cap \\tau \\ni U_{i} \\implies U_{i} \\in \\tau\n\\implies \\cap_{i \\in F} \\; U_{i} \\in \\tau \\implies \\cap U_{i} \\in \\cap_{F \\in \\tau} \\;\\tau\nx_{i} \\to x\n\\exists j such that x_{i} = x, \\; i \\geq j.\n\n"},"Definitions/Topological-Spaces/Terminologies/Compact":{"title":"Compact","links":["Definitions/Sets/Open-Cover","Definitions/Subcover"],"tags":[],"content":"Definition\nX is compact if every open cover has a finite subcover."},"Definitions/Topological-Spaces/Terminologies/Connected-Component":{"title":"Connected Component","links":["Definitions/Topological-Spaces/Topological-Space","Definitions/Topological-Spaces/Terminologies/Connected"],"tags":[],"content":"Definition\nA connected component of a topological space is the union of all connected subsets that contain a given point. It itself is connected."},"Definitions/Topological-Spaces/Terminologies/Connected":{"title":"Connected","links":["Definitions/Topological-Spaces/Topological-Space","Definitions/Sets/Open-Sets"],"tags":[],"content":"Definition\nA topological space is connected if it is not a union of two non-empty open sets.\ni.e. if you draw the two non-empty open sets on the graph, if you have to lift your pen, it will not be connected.\nExamples\nNot connected:\n\\langle 0, 1 ] \\cup [ 2, 5 \\rangle\nConnected:\n\\langle 0, 1 ] \\cup [ 0.5, 5 \\rangle = \\langle 0, 5 \\rangle"},"Definitions/Topological-Spaces/Topological-Space":{"title":"Topological Space","links":["Definitions/Sets/Open-Sets","Definitions/Topological-Spaces/Terminologies/Compact","Definitions/Sets/Open-Cover","Definitions/Subcover"],"tags":[],"content":"Definition\nLet X be a topological space\n\nA neighbourhood of x \\in X is an open set A with x \\in A.\nX is Hausdorff if for x \\neq y\\; \\exists \\, \\text{neighbourhoods} \\, A, B of x and y respectively, such that A \\cap B = \\emptyset.\nA \\subset X is closed if A^{\\complement} is open.\nThe closure (denoted by a bar over the set) \\overline{Y} of a subset Y \\subset X is the intersection of all closed subsets of X that contain Y.\nX is compact if every open cover has a finite subcover.\nX is locally compact if any x \\in X has a neighbourhood with compact closure.\nX is \\sigma-compact if it is a countable union of compact subsets with respect to the relative topology, i.e. an open set of a subset Z of X is of the type Z \\cap A where A is open in X.\n\nDense\nSay Y is dense in X if \\overline{Y} = X. If Y is countable.\nSeparable\nSay \\overline{Y} = X, then we say that X is separable."},"Definitions/Topological-Spaces/Topology":{"title":"Topology","links":["Definitions/Sets/Open-Sets","Definitions/Metric-Spaces/Ball"],"tags":[],"content":"Definition\nA collection of subsets of X, called open sets, such that:\n\nX, \\, \\emptyset \\in \\tau\nAny union of sets from \\tau will be in \\tau.\n\n\n\n \n Info\n \n \n\ny, z \\in \\tau \\implies y \\cup z \\in \\tau\nx_{i} \\in \\tau \\implies \\cup_{i \\in I} X_{i} \\in \\tau\n\n\n\nAny finite intersection of sets from \\tau will be in \\tau.\n\n\n\n \n Info\n \n \n\ny \\cap z \\in \\tau\n\\cap_{i \\in I} X_{i} \\notin \\tau\n\n\nExamples\n\n\n \n Example\n \n \n\nThe topology induced by a metric on X is the collection of all unions of balls.\n\n\n\n\n \n Reasoning for having point/rule 3 in Definition\n \n \n\nConsider the topology on \\mathbb{R} induced by the usual distance.\nB_{r}(x) = \\langle x - r, x + r \\rangle\nNote:\n\\cap_{n \\in \\mathbb{N}} \\langle -\\frac{1}{n}, \\frac{1}{n} \\rangle = \\{ 0\\}\n(\\cap_{n \\in \\mathbb{N}} is an infinite intersection of all numbers (in \\mathbb{N}))\nBut the reason why this is not = \\{ 0, \\varepsilon \\} is a finite amount of intersections\n\\varepsilon \\notin \\langle -\\frac{1}{n}, \\frac{1}{n} \\rangle for n \\gt \\frac{1}{\\varepsilon}\n\n"},"Definitions/Topological-Spaces/Tychonoff-Theorem":{"title":"Tychonoff Theorem","links":["Definitions/Topological-Spaces/Induced/Product-Topology"],"tags":[],"content":"Can be abbreviated as “THM”.\nDefinition\nThe product of compact spaces is compact in the product topology"},"Definitions/Vector-Spaces/Complex-Vector-Space":{"title":"Complex Vector Space","links":["Definitions/Vector-Spaces/Properties-of-a-Vector-Space"],"tags":[],"content":"Definition\nA complex vector space is a set V with addition u + v of vectors u, v, and scalar multiplication a \\times v, \\; a \\in \\mathbb{C}, \\; v \\in V, satisfying the Properties of a Vector Space"},"Definitions/Vector-Spaces/Linear-Basis":{"title":"Linear Basis","links":[],"tags":[],"content":"Definition\nA linear basis of V is a subset \\{ v_{i} \\} \\subset V such that every vector can be written uniquely as \\Sigma_{i} c_{i} v_{i} for finitely many non-zero c_{i} \\in \\mathbb{C}."},"Definitions/Vector-Spaces/Normed-Vector-Space":{"title":"Normed Vector Space","links":["Definitions/Norm"],"tags":[],"content":"Definition\nHave a metric given by the Norm on a vector space V as d(u, v) \\equiv \\|u - v\\| \\in [ \\, 0, \\infty \\rangle."},"Definitions/Vector-Spaces/Properties-of-a-Vector-Space":{"title":"Properties of a Vector Space","links":[],"tags":[],"content":"\nu + v = v + u,\n(u+v) + w = u + (v + w),\nu + 0 = u,\nu + (-u) = 0,\na (u + v) = a \\times u + a \\times v,\n(a + b) \\times v = av + bv,\na(bv) = (ab) \\times v,\n1 \\times v = v.\n\nThese can be used for Complex, Real, or Rational vector spaces."},"Lectures/Lecture-1---1.1-Sets-and-Numbers":{"title":"Lecture 1 - 1.1 Sets and Numbers","links":["Definitions/Terminology/QED","Definitions/Statements/And","Definitions/Statements/Or","Definitions/Statements/Not","Definitions/Statements/Implies"],"tags":[],"content":"Prime Numbers\n\\mathbb{N} = \\{1,2,3,\\dots\\} (natural numbers)\nPrime Number - p \\in \\mathbb{N} \\setminus \\{1\\} only divisible by 1 and p.\nThey are building blocks for multiplication; for instance\n90 = 9 \\times 10 = 3 \\times 3 \\times 2 \\times 5 = 2 \\times 3^2 \\times 5 = 5 \\times 3^2 \\times 2\np \\times q = p' \\times q'\n\\implies (need proof Theorem 1.1.1)\np = p' \\; \\cap \\; q=q'\np = q' \\; \\cap \\; q = p'\nTheorem 1.1.1\nAny natural number other than one is a product of unique primes\nProof Existence\nDivide as long as possible\nUniqueness (Gauss): Need Euclid’s lemma, saying that\nIf n|ab with gcd(a,b) = 1, then n|a or n|b.\nThis lemma follows from the axiom:\nEach non-empty subset of \\mathbb{N} has a least element/number.\nQED\nCOR 1.1.2\nThere are infinitely many primes.\nProof\nSay we had finitely many primes p_{1}, \\ldots, p_{n}. Applying Theorem 1.1.1 to p_{1} \\times p_{2} \\times \\ldots \\times p_{n} + 1 gives the absurdity that 1 can be divided by some prime number\nThis is impossible as for example p_{1} \\times p_{2} \\times \\ldots \\times p_{n} + 1 and p_{1} \\times p_{7} \\times p_{n}, you can divide both sides by something like p_{1}, however on the LHS with + 1 would result in + 1 \\frac{1}{p_{1}}\nQED\nStatements\nThese are mostly similar to logic in computer science with And, Or, Not, and Implies.\nSets\nA set X is characterised by its elements or members x \\in X.\nThey can be listed like \\{1,5,4\\}, or described by some property, like the set of all primes, or like X = \\{x | P(x)\\}; here x is from the outset supposed to belong to some (universal) set. Otherwise X = \\{ x | \\notin X \\} (Russel’s paradox) - which is not allowed. X = \\{ x \\in | x > 7\\} - which is OK!\nY \\subset means x \\in Y \\implies x \\in X\nGet \\emptyset \\subset X\nUnion\nThe union \\cup_{i \\in I} X_{i} consists of x \\in X_{i} for at least one i \\in I\nDisjoint union when X_{i} \\cap X_{j} = \\emptyset for all possible i and j.\nIntersection\nThe intersection \\cap_{i \\in I} X_{i} consists of x \\in X_{i}, \\; \\forall i \\in I\nComplement\nThe complement X \\setminus Y of Y in X consists of x \\in X \\cap x \\notin Y\nWrite Y^\\complement (a complement of Y) when X is understood.\nProduct\nThe product X \\times Y consists of the ordered pairs\n(x, y) \\neq (y, x) \\equiv \\{ \\{ y \\}, \\{ y, x \\} \\}\n(x \\in X and y \\in Y)\n(x,y) = (x', y') \\iff x = x' \\cap y = y'\nA more compact way of writing this: X \\times Y = \\{ (x, y) | x \\in X \\cap y \\in Y \\}\nRelation\nA relation on a set X is R \\subset X \\times X with xRy \\equiv ((x,y) \\in R). (R here meaning is related to)\nExample\nR= \\{ (x, x) | x \\in X \\} \\subset X \\times X, xRy \\iff (x, y) \\in R \\implies x = y.\nThese two elements x and y can only relate if they are the same.\nEquivalence Relation\nAn equivalence relation 0 \\sim X is a relation on \\sim on X such that:\n\nx \\sim x\nx \\sim y \\implies y \\sim x\n(x \\sim y) \\cap (y \\sim z) \\implies x \\sim z\nFor all x,y,z \\in X.\n\nIt partitions X into a disjoint union \\frac{X}{\\sim} of equivalence classes [x] \\equiv \\{ y \\in X | y \\sim x \\}, with x called a representative of [x] (equivalence class)."},"Lectures/Lecture-11":{"title":"Lecture 11","links":["Definitions/Nets","Definitions/Topological-Spaces/Topological-Space","Definitions/Topological-Spaces/Hausdorff","Definitions/Subnet","Definitions/Topological-Spaces/Terminologies/Compact","Definitions/Cauchy-Sequence","Definitions/Sets/Open-Sets","Definitions/Topological-Spaces/Terminologies/Connected","Definitions/Terminology/Bijective","Definitions/Topological-Spaces/Topology"],"tags":[],"content":"Last lecture talked about Nets\nProposition\nA topological space is Hausdorff \\iff each net converges to at most one point.\nProof\n\\Rightarrow:\n“Easy”\n\n\\Leftarrow:\nsay x \\neq y, therefore cannot be separated by disjoint neighbourhoods.\nBy the axiom of choice, pick x_{(A, B)} \\in A \\cap B, where A and B are neighbourhoods of x and y respectively. Consider the index set of pairs (A, B) with (A, B) \\geq (A', B') if A \\subset A' \\cap B \\subset B'.\nThis is a “ufos”, and \\{x_{(A, B)} \\to x; \\; x_{(A, B)} \\to y which is a contradiction.\n\n\\underbrace{x_{(A,B)}}_{\\in {A \\cap B}} \\in A' \\forall \\underbrace{(A, B) \\geq (A', B')}_{\\implies A \\subset A' \\cap B \\subset B'}\nProposition - Convergence in Topological Space\nf: X \\to Y topological spaces with x \\in X.\nThen:\nf is continuous at x \\iff f(x_{i}) \\to f(x) \\; \\forall x_{i} \\to x.\n(\\forall neighbourhood B of f(x) \\exists neighbourhood A of x such that f(A) \\overbrace{\\subset}^{A \\subset f^{-1}(B)} B )\nProof\n\\Rightarrow:\nSuppose x_{i} \\to x and that B is a neighbourhood of f(x). Then there \\exists a neighbourhood A of x such that f(A) \\subset B. Then \\{ x_{i} \\} (net) will eventually be in A. Then \\{ f(x_{i}) \\} will eventually be in B, so that means f(x_{i}) \\to f(x).\n\\Leftarrow:\n(Going for a proof by contradiction)\nSay B is a neighbourhood of f(x) such that every neighbourhood A of x intersects X \\setminus f^{-1}(B). Then x belongs to the closure of X \\setminus f^{-1}(B). By previous proposition there \\exists net \\{ x_{i} \\} \\subset X \\setminus f^{-1} (B) such that x_{i} \\to x.\n\n\\text{to }x \\; \\text{circle}\\iff x \\in \\overline{A} \\iff x \\leftarrow x_{i} \\in A\nDefinition\nA subnet of a net f: I \\to X is a net g: J \\to X and a map h : J \\to I such that g = f \\circ h and such that \\forall i \\in I \\exists j \\in J with h(j') \\geq i \\; \\forall j' \\geq j.\nTheorem\nA space is compact \\iff every net has a converging subnet.\nExamples\n\n\n \n Example: Illustrates \\Rightarrow\n \n \n\nx_{n} = n\n\nx_{n} = \\frac{1}{n} \\in \\langle \\, 0, 1 \\, ]\n\n\n\n\n \n Example: Illustrates \\Rightarrow\n \n \n\nConsider the compact space [ \\, 0, \\, 1 \\,], say \\{ y_{n} \\subset [\\, 0, \\, 1, \\, ] \\}\n\nConstruct subsequence:\nx_{1} = y_{1} x_{2} any y_{j} in the half with infinitely many y_{i}s.\nPick x_{3} to be in the half with infinitely many y_{i}s.\nGet subsequence \\{ x_{n} \\} contained in more and more narrow intervals. So \\{ x_{n} \\} will be cauchy incomplete [ \\, 0, \\, 1 \\, ], so it converges.\n\n\n \n \\underbrace{[\\, 0, \\, 1 \\, ]}_{\\text{closed} \\implies complete} \\subset \\underbrace{\\mathbb{R}}_{\\to \\, \\text{complete}}\n \n \n\n\n\nExercises\n\n\n \n Note\n \n \n\nQuestion numbering is probably not the same as the ones from the exercises on Canvas. The numbering was done in order they appeared in the lecture.\n\n\nQuestion 1\nf, \\, g \\; \\text{contiuous} \\implies f \\circ g \\; \\text{continuous}\nX \\xrightarrow{g} Y \\xrightarrow{f} Z and X \\xrightarrow{(f \\circ g)(x) = f(g(x))} Z\nf^{-1}(A) open in Y\nfor A open in Z.\ng^{-1}(B) open in X\nfor B open in Y\nIs (f \\circ g)^{-1}(A) open in X when A is open in Z?\n(f \\circ g)^{-1}(A) = g^{-1}(\\underbrace{f^{-1}(A)}_{B})\nTake A open in Z. Then (f \\circ g)^{-1}(A) = g^{-1}(f^{-1}(A)) \\xleftarrow{\\text{claim}} \\text{true for any subset of } A \\text{ of } Z is open in X since B = f^{-1}(A) is open in Y, as f is continuous.\nBut then g^{-1}(B) is open in X as g is continuous. Hence (f \\circ g)^{-1}(A) is open in X, so f \\circ g is continuous.\nClaim:\n(f \\circ g)^{-1}(A) = g^{-1}(f^{-1}(A)), \\; \\forall A \\subset Z.\nProof:\nSay x \\in (f \\circ g)^{-1}(A), so (f \\circ g)(x) \\in A, or f(g(x)) \\in A, so g(x) \\in f^{-1}(A), so x \\in g^{-1}(f^{-1}(A)).\nIf x \\in g^{-1}(f^{-1}(A)), then g(x) \\in f^{-1}(A), so f(g(x)) \\in A = (f \\circ g)(x), so x \\in (f \\circ g)^{-1}(A).\nAlternatively:\nSay we have x_{i} \\to x in X. Then (f \\circ g)(x) \\overbrace{=}^{\\text{Definition of } \\circ} f(g(x)) \\overbrace{=}^{x_{i} \\to x} f(g(\\lim_{ i \\to \\infty }x_{i})) (Note: I’m assuming the \\lim is i \\to \\infty, as it was not defined in the lecture) = f(\\lim_{ i \\to \\infty }g(x_{i})) = \\lim_{ i \\to \\infty }f(g(x_{i})) = \\lim_{ i \\to \\infty }(f \\circ g)(x_{i}).\nQuestion 2\nX \\simeq Y\n\\overbrace{S}^{\\text{Compact}} \\not\\simeq \\overbrace{\\mathbb{R}}^{\\text{Not compact}}\n\\mathbb{R}^2\n\\mathbb{R} \\to \\mathbb{R}^2\nx \\mapsto (x, x)\n\\underbrace{\\mathbb{R}}_{\\underbrace{\\simeq}_{\\tan} \\underbrace{\\langle \\, 0, \\, 1 \\, \\rangle}_{\\simeq \\langle \\, -\\frac{\\pi}{2}, \\, \\frac{\\pi}{2} \\, \\rangle}} \\to S \\subset \\mathbb{R}^2\n\\langle \\, -\\frac{\\pi}{2}, \\, \\frac{\\pi}{2} \\, \\rangle \\xrightarrow{\\tan} \\mathbb{R} = \\langle \\, - \\infty, \\, \\infty \\, \\rangle\nS \\not\\simeq \\mathbb{R}^2 by compactness argument.\n\\mathbb{R}^2 is ‘more connected’ than \\mathbb{R}.\nQuestion 3\nWhy does there not exist a continuous injection for S \\to \\mathbb{R}?\nWhat happens then if you take the image f : S \\to \\mathbb{R}?\nf(S) \\subset \\mathbb{R} is both compact and connected, so f(S) = [ \\, a, \\, b \\, ] for some a, \\, b \\in \\mathbb{R}.\nThen f : S \\to [ \\, a, \\, b \\, ] is continuous and bijective (as nothing is excluded from the image).\nSince both spaces (Note: not sure if it is the correct link) are compact and Hausdorff, then f^{-1} is also continuous, so S \\simeq [ \\, a, \\, b \\, ]. But removing one point leaves S connected, but not [ \\, a, \\, b \\, ]. So this is a contradiction.\nQuestion 4\nShow that the connected components of \\mathbb{Q} \\subset \\mathbb{R} consists of single points, and that one of these are open.\nTopology on \\mathbb{Q} = \\{ \\mathbb{Q} \\cap A \\mid A \\text{ open in } \\mathbb{R} \\}\n\nA \\cap Q\n\\{ p \\}, \\; p \\in \\mathbb{Q}\n\\mathbb{Q} \\cap \\langle - \\varepsilon + p, \\, \\varepsilon + p \\rangle contains more points than p\n\nIn the graph above: a \\in \\mathbb{R} \\setminus \\mathbb{Q}."},"Lectures/Lecture-12---Induced-Topologies":{"title":"Lecture 12 - Induced Topologies","links":["Definitions/Topological-Spaces/Induced/Weakest-Topology","Definitions/Topological-Spaces/Induced/Initial-Topology","Definitions/Nets","Definitions/Sets/Open-Sets","Definitions/Topological-Spaces/Topological-Space","Definitions/Topological-Spaces/Continuous","Definitions/Topological-Spaces/Induced/Product-Topology","Definitions/Topological-Spaces/Tychonoff-Theorem","Definitions/Topological-Spaces/Induced/Separating-Points","Definitions/Topological-Spaces/Hausdorff"],"tags":[],"content":"Weakest Topology\nDefinition of Weakest Topology defined in the lecture.\nInitial Topology\nDefinition of Initial Topology, defined in the lecture.\nProposition\nLet X be a set with initial topology induced by a family, F, of functions.\nThen a net in \\{ x_{i} \\} in X converges to x \\iff f(x_{i}) \\to f(x) \\; \\forall f \\in F.\nProof\n\\Rightarrow:\nIs obvious.\n\\Leftarrow:\nLet A be a neighbourhood of x.\nHence there are finitely many open sets A_{n} \\subset Y_{f_{n}} such that x \\in \\cap f^{-1}_{n}(A_{n}) \\subset A. Then A_{n} is a neighbourhood of f_{n}(x), \\; \\forall n.\nBy assumption f_{n}(x_{i}) \\to f_{n}(x), so f_{n}(x_{i}) \\in A_{n} \\; \\forall i \\geq i(n) for some i(n). Pick j such that j \\geq i(n) for all the finitely many ns.\nThen x_{i} \\in \\cap f^{-1}_{n}(A_{n}) \\subset A for all i \\geq j, so x_i \\to x.\nQED.\nCOR\nX has initial topology induced by F. Say Z is a topological space, then:\ng : Z \\to X is continuous f \\circ g is continuous \\forall f \\in F.\nProof\n\\Rightarrow:\nClear.\n\\Leftarrow:\nSay we have a net \\{ z_{i} \\} that z_{i} \\to z in Z.\nThen \\underbrace{(f \\circ g)(z_{i})}_{= f(g(z_{i}))} \\to \\underbrace{(f \\circ g)(z)}_{f(g(z))} \\; \\forall f \\in F.\nHence g(z_{i}) \\to g(z) by the proposition, so g is continuous.\nProduct Topology\nDefinition of the Product Topology, defined in the lecture.\n\n(x_{i}, y_{i}) \\to (x,y)\nBy the previous proposition a net \\{ x_{i} \\} in \\Pi X_{\\lambda} converges to x with respect to the product topology \\iff \\pi_{\\lambda} \\to \\pi_{\\lambda}(x), \\; \\forall \\lambda.\nNote\n\\pi_{\\lambda} are continuous (obvious) and open sets \\pi_{\\lambda}(A) open in X_{\\lambda} for A open in \\Pi X_{\\lambda}\nTychonoff\nIt is the Tychonoff Theorem (defined in lecture).\nSeparating Family\nDefines Separating Points from the lecture.\nProposition\nA set X with initial topology induced from a separating family of functions f : X \\to Y_{f} is Hausdorff when all Y_{f} are Hausdorff.\nProof\nSay x \\neq y in X. Then \\exists f such that f(x) \\neq f(y) in Y_{f}.\nCan separate f(x) and f(y) by neighbourhoods U and V such that U \\cap V = \\emptyset.\nThen f^{-1}(U) and f^{-1}(V) will be disjoint neighbourhoods of x and y.\n\n\n \n Reasoning for f^{-1}(U) and f^{-1}(V) being disjoint\n \n \n\nz \\in f^{-1}(U) \\cap f^{-1}(V)\nf(z) \\in U \\cap f(z) \\in V\nWhich is not possible\n\n\nQED.\nCOR\nA product of Hausdorff spaces is Hausdorff in the product topology.\n\\Pi X_{\\lambda}\n\\Pi_{\\lambda}\nf: X \\to Y_{f}\nf : X_{f} \\to Y\nq : X \\to X \\setminus \\sim\nq(x) = [x]"},"Lectures/Lecture-13---Measure-Theory":{"title":"Lecture 13 - Measure Theory","links":["Definitions/Measure-Theory/Sigma-Algebra/Sigma-Algebra","Definitions/Measure-Theory/Sigma-Algebra/Borel-Sets","Definitions/Measure-Theory/Sigma-Algebra/Borel-Measurable","Pointwise","Definitions/Topological-Spaces/Continuous","Definitions/Sets/Open-Sets","Definitions/Terminology/Countable","Definitions/Topological-Spaces/Terminologies/Compact","Definitions/Topological-Spaces/Hausdorff","Definitions/Sets/Open-Cover","Definitions/Subcover","Definitions/Topological-Spaces/Topology","Definitions/Terminology/Bijective","Definitions/Sets/Open-Map","Definitions/Topological-Spaces/Topological-Space"],"tags":[],"content":"Sigma Algebra\nDefined in the lecture: Sigma-Algebra\n(See also Measurable)\nMeasure\n(See Measure)\n\n\n \n Example\n \n \n\nM = \\wp(X) is a \\sigma-algebra with the counting measure \\mu given by\n\\mu(A) = \\begin{cases}|A|, & \\text{when}\\ A\\ \\text{is finite}\\\\ \\infty, & \\text{when}\\ A\\ \\text{is infinite}\\end{cases}\n\\mu'(A) = 0,\\ \\forall A\n\n\nGiven a collection N of subsets of any set X, the intersection M of all \\sigma-algebras containing N is a \\sigma-algebra; the one generated by N.\nBorel Sets\nDefined in the lecture: Borel Sets\n\n\n \n Example of Borel Sets\n \n \n\n\\cap^{\\infty}_{n=1}\\left[ -\\frac{1}{n}, \\frac{1}{n} \\right] = \\{ 0 \\}\n\n\nProposition\nSay X has a \\sigma-algebra. Then all measurable functions X \\to \\mathbb{C} is an algebra under pointwise operations.\nProof\nSay f, g are measurable and a, b \\in \\mathbb{C}.\nNote that k : \\mathbb{C} \\to \\mathbb{C} given by b \\mapsto ab is continuous then (af)(x) = (\\underbrace{k}_{\\text{continuous}} \\circ \\underbrace{f}_{\\text{measurable}})(x) (= k(\\underbrace{f(x)}_{b}) = ab = a \\times f(x) = (af)(x))\nSince \\mathrm{Re}, \\mathrm{Im} : \\mathbb{C} \\to \\mathbb{R} by \\mathrm{Re}(a + ib) = a and \\mathrm{Im}(a + ib) = b are continuous the real and imaginary parts of f are measurable.\nSo to show that f+g, f \\times g are measurable, we may assume that f and g are real.\n\n\n \n Note\n \n \n\nf + g = \\mathrm{Re}(f+g) + i \\mathrm{Im}(f+g) = \\mathrm{Re} f + \\mathrm{Re} g + i (\\mathrm{Im} f + \\mathrm{Im} g)\n(a, b) \\mapsto x + i b is continuous.\n\n\nLet h : x \\to \\mathbb{R}^2; h(x) = (f(x), g(x)).\nClaim that h is measurable.\nThen use that f + g, f \\times g are compositions of h with the continuous functions \\mathbb{R}^2 \\to \\mathbb{R} given by (s, t) \\xmapsto{p} s + t and (s, t) \\xmapsto{q} s \\times t, so p \\circ h = f + g and q \\circ h = f \\times g.\n(All that needs to be proven now is that h is measurable)\nShow the claim: note that open V \\subset \\mathbb{R}^2 is a countable union of rectangles I \\times J for segments I, J \\subset \\mathbb{R}.\nAlso note that (taking the inverse image of the rectangle) h^{-1}(I \\times J) = f^{-1}(I) \\cap g^{-1}(J) is measurable.\nThen so is h^{V} = h^{-1}(\\cup_{n}(I_{n} \\times J_{n})) = \\cup_{n}h^{-1}(I_{n} \\times J_{n}).\nQED.\n\nExercise Part of the session…\nQuestion 3 (Exercise 6)\nShow that a sequence E_{1} \\supset E_{2} \\supset E_{3} \\supset \\dots of a compact non-empty subsets of a Hausdorff space has a non-empty intersection \\cap E_{i} \\neq \\emptyset\nProof\nNet argument\nX = \\mathbb{R}\nE_{n} = \\left[ -\\frac{1}{n}, \\frac{1}{n} \\right]\n[-1,1] \\supset \\left[ -\\frac{1}{2}, \\frac{1}{2} \\right] \\supset \\left[ -\\frac{1}{3}, \\frac{1}{3} \\right] \\supset \\dots\n\\cap_{n} [-\\frac{1}{n}, \\frac{1}{n}] = \\{ \\emptyset \\} \\neq \\emptyset\nI = \\{ E_{i} | i=1,\\dots,\\infty \\}\nUpward filtered ordered set under reverse inclusion.\nDefined by axiom of choice X_{E_{i}} \\in E_{i}, so we get a net \\{ x_{E_{i}} \\}_{E_{i} \\in I} \\subset E_{i}\nThen it has a convergent subnet \\{ x_{{E_{i}}_{j}} \\}_{j}, say x_{{E_{i}}_{j}} \\to x \\in E_{1}.\nClaim x \\in \\cap E_{i}:\nIf A is a neighbourhood of x in E_{1}, then x_{{E_{i}}_{j}} \\in A\\ \\forall j \\geq k.\nSo A \\cap {E_{i}}_{j} \\neq \\emptyset\\ \\forall j \\geq k.\n\nIf x \\notin \\cap E_{i}, then \\exists neighbourhood\nB of x such that B \\cap (\\cap E_{i}) = \\emptyset.\nThen B \\cap E_{i} = \\emptyset\\ \\forall i \\geq m.\nPick B=A and get a contradiction.\nSubcover argument\nV_{n} = E_{n}^{\\complement} open in Hausdorff space if \\cap E_{i} = \\emptyset, then \\{ V_{n} \\} will be an open cover of E_{1} that has no finite subcover, contradicting that E_{1} is compact.\n\\cup V_{n} = \\cup E_{n}^{C} = (\\cap E_{n})^{\\complement} = \\emptyset^{\\complement} = X\nQuestion 2 (Exercise 7)\nShow that compact Hausdorff spaces are rigid; if X is compact Hausdorff it has no weaker or stronger topology that is compact Hausdorff.\nProof\n\\iota : (X, \\overbrace{\\tau}^{\\text{Compact Hausdorff}}) \\to (X, \\overbrace{\\tau}^{\\text{Weaker}}) (\\tau' \\subset \\tau)\n\\iota(x) = x, so \\iota is bijective\nThis is a continuous map since \\tau' \\subset \\tau.\n\\iota is also an open map, because it takes closed sets to closed sets, because if E is closed in (X, \\tau), then it is compact, and \\iota(E) is compact in (X, \\tau') as \\iota is continuous, so \\iota(E) is closed in (X, \\tau') as it is Hausdorff. So in this case \\tau' = \\tau.\nQuestion 1 (Exercise 7)\nA topological space is second countable if we have a sequence \\{ V_{n} \\} of open sets such that any open set is a union of some of these V_{n}s.\nProof\n\\langle c, d \\rangle = \\cup V_{\\frac{1}{n}}^{a}\nV_{n} = \\{ n \\}\n\\mathbb{R} = \\cup_{a \\in \\mathbb{R}}V_{a}\nV_{a} = \\{ a \\}\n(which is not countable)\nInstead:\nI \\times J\nAny open A \\subset \\mathbb{R} countable union of I.\nX is separable if it has a dense sequence \\{ x_{n} \\} = X.\nClaim:\nX selectable countable \\implies X separable\nAxiom of choice def x_{n} \\in V_{n}. Then \\overline{{ x_{n} }} = X. If \\overline{{ x_{n} }} \\neq X$.\nx \\in \\overline{\\{ x_{n} \\}}^{\\complement} \\exists open A with x \\in A and A \\cap \\overline{\\{ x_{n} \\}} = \\emptyset. But \\exists m such that x \\in V_{m}…"},"Lectures/Lecture-2":{"title":"Lecture 2","links":["Definitions/Rational-Cauchy-Sequences"],"tags":[],"content":"Quotients\nConstructing Quotients\nHave equivalence relation on \\mathbb{Z} \\times (\\mathbb{Z} \\setminus \\{ 0 \\}) given (a,b) \\sim (c,d) when ad = bc.\nThen \\frac{\\mathbb{Z} \\times \\mathbb{Z} \\setminus \\{ 0 \\}}{\\sim} is the set of rational numbers \\mathbb{Q} with addition and multiplication defined in an obvious way.\n\\frac{a}{b} = \\frac{c}{d}\nReal Numbers\nConstructing Real Numbers\nProblem with \\mathbb{Q}:\nFor example Pythagoras’ Theorem with sides 1 and 1 gives \\sqrt{ 2 }\n\\sqrt{ 2 } \\notin \\mathbb{Q}\n\\sqrt{ 2 } = \\frac{a}{b}, \\; a,b \\in \\mathbb{Z}, \\; b \\neq 0.\nWhich is a contradiction\nYet we can find a sequence \\{ X_{n} \\} of rational numbers such that X_{n}^2 \\to 2 as n \\to \\infty.\nx_{1} = 1, \\; x_{2} = 1.4, \\; x_{3} = 1.41\n\\sqrt{ 2 } = 1.4142\\ldots\nBy a sequence \\{ x_{n} \\} in a set X we mean a function f : \\mathbb{N} \\to X with x_{n} \\equiv f(n), and that a function or map X \\to Y between two sets ascribes one member of Y to each member of X.\nA sequence \\{ x_{n} \\} in \\mathbb{Q} converges to x \\in \\mathbb{Q}, written \\lim_{ x_{n} \\to \\infty} = x, if \\forall k \\in \\mathbb{N} \\exists N_{k} \\in \\mathbb{N} such that | x - x_{n} | \\lt \\frac{1}{k}, \\; \\forall n \\lt N_{k}.\nSo what is \\sqrt{ 2 }?\nReal numbers are certain equivalence classes of Rational Cauchy Sequences\nTwo such sequences \\{ x_{n} \\}, \\{ y_{n} \\} are equivalent if the “distance” \\lim_{ |x_{n} - y_{n}| \\to \\infty } = 0 between them vanishes.\n\\{ x_{n} \\} \\in X \\sim \\{ y_{n} \\} \\in X\nTheir set of equivalence classes is the set of real numbers \\mathbb{R}.\nX \\setminus \\sim = \\mathbb{R} \\ni [\\{ x_{n} \\}]\n[\\{ x_{n} \\}] + [\\{ y_{n} \\}] = [\\{ x_{n} + y_{n} \\}]\nGet natural algebraic operations on \\mathbb{R} from \\mathbb{Q}; check well-definedness. Then \\mathbb{Q} \\subset \\mathbb{R} as the classes containing constant sequences. An order \\gt on \\mathbb{R} is defined by declaring as positive those classes having sequences with only positive rational numbers.\nx \\gt y \\implies x-y \\gt 0\nConvergence of Real Numbers\nA sequence of real numbers is said to converge to a real number if the “distance” between their representatives tend to zero."},"Lectures/Lecture-3":{"title":"Lecture 3","links":["Definitions/Number-Field","Definitions/Cauchy-Sequence","Definitions/Least-Upper-Bound-Property","Definitions/Terminology/Injective","Definitions/Terminology/Surjective","Definitions/Terminology/Bijective","Definitions/Terminology/Countable","Definitions/Functions/Direct-Product","Definitions/Functions/Power-Set","Definitions/Functions/Characteristic-Function"],"tags":[],"content":"Proposition 1.1.4\nEach real number is a limit of a sequence of rational numbers.\n\\mathbb{Q} \\subset \\mathbb{R}\nAs an ordered Number Field \\mathbb{R} is complete, meaning that every Cauchy Sequence in \\mathbb{R} converges to a real number.\nEquivalently, the real numbers have the Least Upper Bound Property; \\forall X \\subset \\mathbb{R} bounded above has a least upper bound denoted by sup(X) \\in \\mathbb{R}. Eq. an inf(Y) \\in \\mathbb{R} if Y bounded below.\nFunctions and Cardinality\nA function f : X \\to Y is Injective if f(x) = f(y) \\implies x = y.\nSurjective if f(x) = y.\nBijective if it is both injective and surjective.\nThen we write X \\simeq Y.\n|X| = |Y| \\; \\text{(cardinality)}\nSay X is Countable if |X| = |\\mathbb{N}|; this means that the members of X can be listed as a sequence with x_{n} = f(n), where f: \\text{IN} \\to X is some bijection.\nCantor’s Diagonal Argument\nThe real numbers cannot be listed, or they are uncountable.\nIndeed, present a list of the real numbers in \\langle 0, 1 \\rangle written as binary expansions. Then the number that has as its n-th digit, the opposite value to the n-th digit of the n-th number of the list, will never be in the list.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\0101100111…0111111111…0000010000…0101010000……So here the bold numbers going diagonally from the \\ shows that they cannot be countable as they are not the same number.Cantor: 0.0011…\nAxiom of Choice\nAny Direct Product\n\\Pi_{i \\in I} \\, X_{i} \\equiv \\{ x : I \\to \\cup_{i \\in I} X_{i} | x_{i} \\equiv x(i) \\in X_{i} \\}\nis non-empty when all x \\neq \\emptyset.\nAny x \\in \\Pi_{i \\in I} \\, X_{i} is called a choice function.\nThe Power Set \\wp(X) of X consists of all the subsets of X.\n\\exists bijection \\wp(X) \\to \\Pi_{x} \\{ 0, \\, 1 \\} = \\{ f : X \\to \\{ 0, \\, 1 \\} \\} that sends Y \\subset X to its Characteristic Function."},"Lectures/Lecture-4---1.2-Metric-Spaces":{"title":"Lecture 4 - 1.2 Metric Spaces","links":["Definitions/Functions/Inverse-Function","Definitions/Sets/Complex-Numbers","Definitions/Cauchy-Sequence","Definitions/Terminology/Algebraically-Complete","Definitions/Metric-Spaces/Metric-Space","Definitions/Vector-Spaces/Normed-Vector-Space","Definitions/Vector-Spaces/Complex-Vector-Space","Definitions/Vector-Spaces/Linear-Basis","Vector-Space"],"tags":[],"content":"The Inverse Image\nThe Inverse Image uses a Inverse Function f^{-1} (z) of Z \\subset Y written f: x \\to y is f^{-1}(z) \\equiv \\{ x \\in X | f(x) \\in Z \\}.\nComplex Numbers\nIn Complex Numbers, \\mathbb{C} = \\mathbb{R} \\times \\mathbb{R} with usual addition of vectors\nMultiplying Vectors\nMultiply vectors by adding their angles multiplying their lengths.\nz = a + i \\times b = (a, b)\n(b here can be seen as (a, 0) + (0, b) = (a+0, 0+b))\nb = (b, 0) \\implies i \\times b = (0, b)\ni \\times z rotates z 90\\degree counterclockwise.\nProposition\n\\mathbb{C} is complete (cauchy) and For Complex Numbers.\nMetric Spaces\nDefinition\nExample\nDiscrete metric on X; d(x, y) = \\begin{cases}0, & \\text{if}\\ x=y\\\\ 1, & \\text{if}\\ x \\neq y\\end{cases}\nVector Spaces\n\nNormed Vector Spaces\nComplex Vector Spaces\n\nExample\nV = R^n = R \\times \\dots \\times R = \\{ (x_{1}, \\, \\dots, x_{n}) | x_{i} \\in \\mathbb{R} \\}\n(where the length of R \\times \\, \\dots \\times R has n \\mathbb{R}s.)\nV = \\mathbb{R}^2 : (x , \\, y) + (z, w) \\equiv (x + z, \\, y + w)\na \\times (x, \\, y) \\equiv (ax, \\, ay)\nLinear Basis Example\nu = 3v_{1} + 5v_{2} + iv_{3}\n3 v_{1} \\neq 2v_{2}\nc_{1}v_{1} + c_{2}v_{2} = 0 \\implies c_{1} = 0 = c_{2}\n\\implies 0 \\times v_{1} + 0 \\times v_{2} = 0\nContinuing the [[#Vector Spaces#Example]] but for Linear bases\nv_{1} = (1, \\, 0, \\, 0, \\, 0, \\, \\dots)\nv_{2} = (0, \\, 1, \\, 0, \\, 0, \\, \\dots)\nv_{3} = (0, \\, 0, \\, 1, \\, 0, \\, \\dots)\nand the way of writing this would be:\n(x_{1}, \\, \\dots, \\, x_{n}) = \\Sigma^{n}_{i=1} x_{i} \\times v_{i} = 0\n\\implies x_{i} = 0\nProposition\nAny Vector Space V has a Linear Basis, and every basis has the same cardinality referred to as the dim(V) (dimension of V) of V.\nProof\n[[ACIT4330/Lectures/Lecture 3#Axiom of Choice|Lecture 3#Axiom of Choice]]"},"Lectures/Lecture-5":{"title":"Lecture 5","links":["Banach-Space","Definitions/Terminology/QED","Isomorphic","Definitions/Terminology/Bijective","Normed-vector-Space","Isometric","Definitions/Hilbert-Spaces","Definitions/Cauchy-Schwarz-Inequality","Definitions/Inner-Product"],"tags":[],"content":"\n\n \n Info\n \n \n\nA normed space is a Banach Space when the corresponding metric space is complete. Any normed space can be completed to a Banach Space, see the real numbers from \\mathbb{Q}.\n\n\n\n\n \n Example: \\mathbb{C}^n vector space, then\n \n \n\n\n\\| \\, v \\, \\|_{1} \\equiv \\Sigma_{k=1}^{n} |v_{k}|, \\; v=(v_{1}, \\dots, v_{n})\n\\| \\, v \\, \\|_{\\infty} \\equiv \\text{sub}_{k=1,\\dots,n}\\{ |v_{k} | \\}\n\nWhich are norms on \\mathbb{C}^n\n\n\nExample\n\n\n \n Question\n \n \n\nConsider C_{c}(X) \\subset \\Pi_{X}\\mathbb{C} = \\{ f : X \\to C \\} on functions X -> \\mathbb{C} that are non-zero for finitely only many x \\in X.\n\n\nThen C_{c}(X) is a vector space under op.\nf \\neq \\text{only on} \\, Y \\, \\subset \\, X\nq \\neq \\, \\text{only on} \\, Z \\subset X\n\\implies f+q \\neq 0 \\, \\text{only on} \\, Y \\cup Z\na \\times f \\neq \\, \\text{only on} \\, Y\nwith \\delta_{x}(y) = \\begin{cases}1, & x=y\\\\ 0, & x\\neq y\\end{cases}\nHas linear basis \\{ \\delta_{x} \\}, x \\in X\nProof\nGiven that f \\in C_{c}(X) then\nf = \\Sigma_{x \\in X} f(x) \\times \\delta_{x} is a finite sum, and the only possibility.\nQED\nSo \\text{dim} C_{c} (X) = |X|.\n\nDefinite norms on C_{c}(X) by \\| \\, f \\, \\|_{1} = \\Sigma_{x \\in X} | f(x) | and \\| \\, f \\, \\|_{\\infty} = \\text{sup}_{x \\in X} | f(x) | when |X| = n we recover \\mathbb{C}^n.\n\nWe write V \\simeq Wand say that V and W are Isomorphic\n\nAs sets if \\exists bijection f : V \\to W\nAs vector spaces if in addition f is linear; f(a \\times u + b \\times v) = a \\times f(u) + b \\times f(v), \\; \\forall a,b \\in \\mathbb{C}, \\; u,v \\in V\nAs normed spaces if in addition f is Isometric; \\| \\, f(v) \\, \\| \\, = \\| \\, v \\, \\|, \\; \\forall v \\in V.\nf should preserve all relevant structure.\n\nHilbert Spaces\nTriangle Inequality\nFollows from the Cauchy-Schwarz Inequality\n\n\n \n Example on C_{c}(X) then\n \n \n\n(f | g) = \\Sigma_{x \\in X} f(x) \\times \\overline{g(x)}\nThis defines an inner product, which can be completed to a Hilbert space.\n\\| \\, f \\, \\|_{2} = (\\Sigma_{x \\in X} | f(x) |^2)^{\\frac{1}{2}}\n\n\n \n The inner product here\n\\mathbb{C}^{n} (u | v) = \\Sigma_{k = 1}^{n} u_{k} \\overline{v_{k}}\n\\mathbb{R}^{n} \\| \\, u \\, \\|_{2} = (\\Sigma |u_{k}|^{2})^{\\frac{1}{2}}\n\\vec{u} \\times \\vec{v} = (\\vec{u} | \\vec{v})\n \n \n\n\n"},"Lectures/Lecture-6---2.1-Topology":{"title":"Lecture 6 - 2.1 Topology","links":["Definitions/Metric-Spaces/Ball","Definitions/Sets/Open-Sets","Definitions/Topological-Spaces/Topological-Space","Definitions/Topological-Spaces/Topology","Definitions/Topological-Spaces/Hausdorff"],"tags":[],"content":"Open Sets\nThe concept of balls is required to understand what an open sets are.\nDefinition\nThe definition can be found in Open Sets.\nIn other words ( ”):” ), for any open \\overbrace{A}^{\\in x} \\exists N such that x_{n} \\in A \\; \\forall n \\gt N.\nIn fact, open sets are unions of balls.\nTopological Space\nA Topological Space X is a set with a Topology \\tau.\n\n\n \n Trivial Topology\n \n \n\nX is the set. \\tau = \\{ \\emptyset, X \\}.\nTherefore x \\neq y, then the only neighbour is X\nX \\cap X = X.\n\n\n\n\n \n Discrete Topology\n \n \n\nX is the set. \\tau = \\wp(X).\n\\{ x \\} \\in \\tau\nX \\, \\text{compact} \\iff X \\, \\text{finite}\nThis topological space is always Hausdorff as it includes all the points on X are included\n\n\n \n A = \\{ x \\}, \\; B = \\{ y \\} \\; A \\cup B = \\emptyset\n \n \n\nd(x, y) = \\begin{cases} 1, & \\text{if}\\ x \\neq y\\\\ 0, & \\text{if} \\, x = y\\end{cases}\n\n"},"Lectures/Lecture-7":{"title":"Lecture 7","links":["Definitions/Sets/Open-Sets","Definitions/Topological-Spaces/Topological-Space","Definitions/Topological-Spaces/Hausdorff","Definitions/Topological-Spaces/Terminologies/Compact","Definitions/Sets/Open-Cover","Definitions/Subcover","Definitions/Terminology/Bounded","Definitions/Cauchy-Sequence","Definitions/Terminology/Bijective","Definitions/Linear-Map"],"tags":[],"content":"(X, d) ball topology\nSay Y is a closed set in X (in other words, Y^{\\complement} open).\nSay we have a sequence \\{ X_{n} \\} in Y and that x_{n} \\to x \\in X.\nThen x \\in Y.\n\n\n \n Proof \n \n \n\nLet’s say x \\notin Y.\n\nWhich is a contradiction.\nBecause x cannot converge outside Y\nQED.\n\n\n(Y, d) complete if (X, d) is complete.\nProposition\nSay X is a topological space, that is Hausdorff.\nIf B \\subset X is compact, then B is closed.\nProof\n\nHausdorff ⇒ A_{y} \\cap B_{y} = \\emptyset , both are open.\nThen \\{ B_{y} \\}_{y \\in B} is an open cover of B\nCompact ⇒ Pick out finite subcover \\{ B_{y_{i}} \\}_{i=1}^{n} of B.\nThen x \\in \\overbrace{\\cap_{i=1}^{n} A_{y_{i}}}^{\\text{Open}} \\subset B^{\\complement} since \\cap_{i=1}^{n} A_{y_{i}} \\cap B_{y_{j}} = \\emptyset (for any j)\nSo B^{\\complement} is open, in other words, B is closed.\nQED.\nHeine-Borel Theorem\nSay A \\subset \\mathbb{R}^n. Then A is compact \\iff A is closed and bounded.\nProof\nFor →: A is closed since \\mathbb{R^n} is Hausdorff and A is compact, see the previous result.\nIt is bounded since we can cover it by finitely many balls.\nFor ←: Assume first that A is an n-cube with boundary included.\nSay A is not compact. If an open cover of A has no finite subcover, then by halving sides of cubes we get a sequence of cubes contained in each other, each having no finite subcover. The centres of these cubes form a Cauchy sequence with a limit x \\in A.\n\nd(x_{n}, x_{m}) \\lt \\varepsilon, \\; \\forall n,\\,m \\gt N. Show: \\mathbb{R}^n is complete.\nAny neighbourhood of x from the cover will obviously contain a small enough cube, and will be a finite subcover.\nBut this is a contradiction.\nQED.\nA^{\\complement} open. Say \\{ \\cup_{n} \\} is an cover of A. Then \\{ \\cup_{n} \\}, \\, A^{\\complement} is an open cover of the n-cube.\n\\{ \\cup_{n_{i}} \\}_{i=1}^N, \\, A^{\\complement} cover of the n-cube.\nThen \\{ \\cup_{n_{i}} \\}_{i=1}^N will cover A.\n\nCor\n\\mathbb{R}^n is locally compact Hausdorff, and \\sigma-compact\n\n\nComplex Vector Space Exercise\nShow that finite dimensional the complex vector space V \\implies V \\simeq \\mathbb{C}^n for some n.\nx,y \\in V\n\\implies x + y \\in V\na \\times x \\in V, \\; a \\in \\mathbb{C}\n\\mathbb{C}^2:\n(x, y) + (z, w) \\equiv (x+z, y+w)\na \\times (x, y) \\equiv (a x, a y)\nLinear map:\nProof\nn = \\dim V. Say \\{ v_{i} \\}_{i=1}^n is a linear basis for V. Define bijective linear map L: \\mathbb{C}^n \\to V by L((x_{1}, \\dots, x_{n})) = \\Sigma_{i=1}^n x_{i} \\times v_{i}. It is linear;\nL(a(x_{1}, \\dots, x_{n}) + b(y_{1}, \\dots, y_{n})) = L((ax_{1} + by_{1}, \\dots, ax_{n} + by_{n})) = \\Sigma_{i=1}^n(ax_{i} + by_{i})v_{i} =\n\\Sigma_{i=1}^n(ax_{i}v_{i} + by_{i}v_{i}) = a \\times \\Sigma_{i=1}^n x_{i}v_{i} + b \\times \\Sigma_{i=1}^ny_{i}v_{i} =\na \\times L((x_{1},\\dots,x_{n})) + b \\times L((y_{1},\\dots,y_{n}))\nSurs.\nv \\in V. Basis \\implies \\exists \\underbrace{x_{i}}_{\\in \\mathbb{C}} such that v = \\Sigma_{i=1}^n x_{i} v_{i} = L((x_{1}, \\dots, x_{n})).\nSo v \\in \\text{lm} L\nInjective\nSay L((x_{1}, \\dots, x_{n})) = L((y_{1}, \\dots, y_{n})) then\n\\Sigma_{i=1}^n x_{i} v_{i} = \\Sigma_{i=1}^n y_{i} v_{i} \\xrightarrow{basis} x_{i} = y_{i}, \\; \\forall i, so x = y\nInjective for linear map L \\iff \\ker L = \\{ 0 \\} \\equiv \\{ x \\in \\mathbb{C} | L(x) = 0 \\}.\n\\ker L is a vector space\n\\text{lm} \\,L is a vector space\nL = 0 \\to \\ker L = \\mathbb{C}^n\n0 \\in V (vectors spaces have to start from the origin, as that is where vectors themselves start from).\nMetric Space Exercise\nConsider the unit circle \\mathbb{T} = S^1 = \\{ (x,y) \\in \\mathbb{R}^2 | x^2 + y^2 = 1 \\}\n\\mathbb{T} = \\{ z \\in \\mathbb{C} \\, | \\, |\\, z \\,| = 1 \\}\nSay there is a circle z = (x, y), x^2 + y^2 = 1\nThis is a metric space for d(z, z') = arclength between z and z'.\nCheck:\n\nd(z', z) = d(z, z')\nd(z, z') = 0 \\iff z = z'\nd(z, z'') \\leq d(z, z') + d(z',z'')\nd'(z, z') = length of straight line between z and z' = \\sqrt{ (x-x')^2 + (y - y')^2 } = \\| \\, z - z' \\, \\|\n"},"Lectures/Lecture-8":{"title":"Lecture 8","links":["Definitions/Topological-Spaces/Topological-Space","Definitions/Topological-Spaces/Terminologies/Connected","Definitions/Topological-Spaces/Terminologies/Connected-Component","connected","Definitions/Sets/Open-Sets","Definitions/Topological-Spaces/Continuous","Definitions/Topological-Spaces/Terminologies/Compact","Inverse-Images","Definitions/Sets/Open-Cover","Heine–Borel","Topological-Invariants"],"tags":[],"content":"Defines in topological spaces:\n\nConnected\nConnected Component\n\nProposition\n[0, 1] is connected.\nProof\n[0, 1] = A \\cup B\nA \\cap B = \\emptyset\nSuppose open sets A and B disconnected it, and say 1 \\in B.\nThen every neighbourhood of a \\equiv \\sup{A} \\in [0, 1] will intersect both A and B.\nWhich is a contradiction as this is impossible!\nQED.\n2.2 Continuity\nRecall: f: \\mathbb{R} \\to \\mathbb{R} is continuous at x if \\lim_{ y \\to x }f(y) = f(x), and it would be discontinuous if \\lim_{ y \\to x } f(y) \\neq f(x). More precisely, if \\forall \\varepsilon \\gt 0 \\, \\exists \\delta \\gt 0 such that |x-y| \\lt \\delta \\implies |f(x) - f(y) < \\varepsilon or f(B_{\\delta}(x)) \\subset B_{\\varepsilon}(f(x)), or B_{\\delta}(x) \\subset f^{-1}(B_{\\varepsilon}(f(x))) = \\{ y \\in \\mathbb{R} | f(y) \\in B_{\\varepsilon}(f(x)) \\}.\nDefinition\nThe definition is defined at: Continuous.\nProposition\nA continuous f : X \\to \\mathbb{R} with X compact, it will attain both a maximum and a minimum.\nProof\nNote: the continuous image of a compact set is compact since inverse images of an open cover will again be an open cover. Then the final step is to use the Heine–Borel theorem since f(x) is compact in \\mathbb{R}.\nQED.\nSomething else\nWe say f is continuous (at every x) if f^{-1}(A) is open for every open A \\subset Y.\nContinuous images of connected spaces are connected. Again because inverse images of open subsets disconnecting an image would disconnect the domain.\nSo homeomorphisms preserve compactness and connectedness. They are topological invariants."},"index":{"title":"ACIT4330 Table of Contents","links":["Lectures/Lecture-1---1.1-Sets-and-Numbers","rnote/ACIT4330-2025-01-06-Lecture-1.rnote","Lectures/Lecture-2","rnote/ACIT4330-2025-01-09-Lecture-2.rnote","Lectures/Lecture-3","Lectures/Lecture-4---1.2-Metric-Spaces","Lectures/Lecture-5","Lectures/Lecture-6---2.1-Topology","Lectures/Lecture-7","rnote/ACIT4330-2025-01-30-Lecture-7.rnote","Lectures/Lecture-8","ACIT4330/Lectures/Lecture-11","Lectures/Lecture-12---Induced-Topologies","Lectures/Lecture-13---Measure-Theory"],"tags":[],"content":"Chapter 1\n1.1 Sets and Numbers\n\nLecture 1 - 1.1 Sets and Numbers (complimentary written notes: ACIT4330-2025-01-06-Lecture 1.rnote)\nLecture 2 (complimentary written notes: ACIT4330-2025-01-09-Lecture 2.rnote)\nLecture 3\nThe Inverse Image and Complex Numbers.\n\n1.2 Metric Spaces\n\nLecture 4 - 1.2 Metric Spaces\nLecture 5\n\nChapter 2\n2.1 Topology\n\nLecture 6 - 2.1 Topology\nLecture 7 (complimentary written notes: ACIT4330-2025-01-30-Lecture 7.rnote)\nLecture 8\n\n2.2 Continuity\n\nLecture 8\nLecture 11\nLecture 12 - Induced Topologies\n\nMeasure Theory\n\nLecture 13 - Measure Theory\n"}} |