Quartz sync: Mar 1, 2025, 2:26 PM

content/Definitions/Topological Spaces/Continuous.md

@@ -0,0 +1,8 @@
+# Definition
+We say $f$ is **continuous** (at every $x$) if $f^{-1}(A) \equiv \{ x \in X | f(x) \in A \}$ is open for every open $A \subset Y$.
+
+We say $f$ is [[Open Sets|open]] if $f(B)$ is [[Open Sets|open]] and $\forall$ [[Open Sets|open]] $B \subset X$.
+
+If $f$ is a [[Bijective|bijection]] that is both **continuous** and [[Open Sets|open]], it is a [[Homeomorphic|homeomorphism]], and $X$ and $Y$ are [[Homeomorphic|homeomorphic]], written $X \simeq Y$; they are the 'same' as [[Topological Space|topological spaces]].
+## In-depth Definition
+A function $f : X \to Y$ between [[Topological Space|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 $B \subset f^{-1}(A)$.

content/Definitions/Topological Spaces/Hausdorff.md

@@ -0,0 +1,4 @@
+# Definition
+If there are two points $x$ and $y$ in a [[Topological Space|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 $U \cup V = \emptyset$.
+
+$X$ is a **Hausdorff space** if any two distinct points in $X$ are separated by neighbourhoods.

content/Definitions/Topological Spaces/Induced/Initial Topology.md

@@ -0,0 +1,4 @@
+# Definition
+The **initial topology** on $X$ induced by a family of functions $f : X \to Y_{f}$ into [[Topological Space|topological spaces]] $Y_{f}$ is the [[Weakest Topology|weakest topology]]on $X$ making all these functions [[Continuous|continuous]].
+
+Here: $F = \{ f^{-1} \; | \; f : X \to Y_{f}, \; A \; \text{open in} \; Y_{f} \}$.

content/Definitions/Topological Spaces/Induced/Product Topology.md

@@ -0,0 +1,13 @@
+# Definition
+The **product topology** on $\Pi X_{\lambda}$, $X_{\lambda}$ [[Topological Space|topological spaces]], is the [[Initial Topology|initial topology]] induced by the family of projections $\Pi_{\lambda}$
+> [!note] What is $\Pi_{\lambda}$?
+> $\pi_{\lambda} : \Pi_{\lambda' \in \wedge} X_{\lambda'} \to X_{\lambda}$
+> $\pi_{\lambda}(f) = f(\lambda)$
+> $\pi_{\lambda}((X_{\lambda'})) = x_{\lambda}$
+
+$$\underbrace{\Pi_{\lambda \in \wedge} X_{\lambda} \equiv}_{\in (x_{\lambda})_{\lambda \in \wedge}} \{ f : \wedge \to \cup_{\lambda \in \wedge} X_{\lambda} \; | \; \underbrace{f(\lambda)}_{x_{\lambda}} \in X_{\lambda} \}$$
+# Example
+Product of 2 [[Topological Space|topological spaces]]: $x_{1} \times x_{2}$
+$= \{ (x_{1}, x_{2}) \; | \; x_{i} \in X_{i} \}$
+$\pi_{1}((x_{1}, x_{2})) = x_{1}$
+![[Drawing 2025-02-24 12.47.32.excalidraw]]

content/Definitions/Topological Spaces/Induced/Separating Points.md

@@ -0,0 +1,2 @@
+# Definition
+A family $F$ of functions on a set **separating points** $x \neq y$ in the set if $f(x) \neq f(y)$ for some $f \in F$

content/Definitions/Topological Spaces/Induced/Weakest Topology.md

@@ -0,0 +1,11 @@
+# Definition
+Given $F \subset \wp(X)$. The **weakest topology** on $X$ that contains $F$ is the intersection of all the [[Topology|topologies]] that contains $F$. This is a [[Topology|topology]], and consists of $\emptyset$, $X$, and all unions of finite intersections of members from $F$.
+
+> [!example]
+> $F \subset \tau$
+> $\textvisiblespace \cap \tau \ni U_{i} \implies U_{i} \in \tau$
+> $\implies \cap_{i \in F} \; U_{i} \in \tau \implies \cap U_{i} \in \cap_{F \in \tau} \;\tau$
+> 
+> $x_{i} \to x$
+> $\exists j$ such that $x_{i} = x, \; i \geq j$.
+

content/Definitions/Topological Spaces/Terminologies/Compact.md

@@ -0,0 +1,2 @@
+# Definition
+$X$ is **compact** if every [[Open Cover|open cover]] has a finite [[Subcover|subcover]].

content/Definitions/Topological Spaces/Terminologies/Connected Component.md

@@ -0,0 +1,2 @@
+# Definition
+A **connected component** of a [[Topological Space|topological space]] is the union of all [[Connected|connected]] subsets that contain a given point. It itself is [[Connected|connected]].

content/Definitions/Topological Spaces/Terminologies/Connected.md

@@ -0,0 +1,10 @@
+# Definition
+A [[Topological Space|topological space]] is **connected** if it is not a union of two non-empty [[Open Sets|open sets]].
+
+i.e. if you draw the two non-empty [[Open Sets|open sets]] on the graph, if you have to lift your pen, it will not be connected.
+
+# Examples
+Not connected:
+$$\langle 0, 1 ]  \cup [ 2, 5 \rangle$$
+Connected:
+$$\langle 0, 1 ] \cup [ 0.5, 5 \rangle = \langle 0, 5 \rangle$$

content/Definitions/Topological Spaces/Topological Space.md

@@ -0,0 +1,13 @@
+# Definition
+Let $X$ be a topological space
+1. A **neighbourhood** of $x \in X$ is an [[Open Sets|open set]] $A$ with $x \in A$.
+2. $X$ is Hausdorff if for $x \neq y\; \exists \, \text{neighbourhoods} \, A, B$ of $x$ and $y$ respectively, such that $A \cap B = \emptyset$.
+3. $A \subset X$ is **closed** if $A^{\complement}$ is open.
+4. The **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 $Y$.
+5. $X$ is **[[Compact|compact]]** if every [[Open Cover|open cover]] has a finite [[Subcover|subcover]].
+6. $X$ is **locally compact** if any $x \in X$ has a neighbourhood with compact closure.
+7. $X$ is $\sigma$-compact if it is a countable union of compact subsets with respect to the **relative topology**, i.e. an [[Open Sets|open set]] of a subset $Z$ of $X$ is of the type $Z \cap A$ where $A$ is open in $X$.
+## Dense
+Say $Y$ is **dense** in $X$ if $\overline{Y} = X$. If $Y$ is countable.
+## Separable
+Say $\overline{Y} = X$, then we say that $X$ is **separable**.

content/Definitions/Topological Spaces/Topology.md

@@ -0,0 +1,25 @@
+# Definition
+A collection of subsets of $X$, called [[Open Sets|open sets]], such that:
+1. $X, \, \emptyset \in \tau$
+2. Any union of sets from $\tau$ will be in $\tau$.
+> [!info]-
+> $$y, z \in \tau \implies y \cup z \in \tau$$
+> $$x_{i} \in \tau \implies \cup_{i \in I} X_{i} \in \tau$$
+3. Any **finite** intersection of sets from $\tau$ will be in $\tau$.
+> [!info]-
+> $$y \cap z \in \tau$$
+> $$\cap_{i \in I} X_{i} \notin \tau$$
+
+# Examples
+> [!example]
+> The **topology induced by a metric** on $X$ is the collection of all unions of [[Ball|balls]].
+
+> [!example] Reasoning for having point/rule 3 in [[#Definition]]
+> Consider the topology on $\mathbb{R}$ induced by the usual distance.
+> $$B_{r}(x) = \langle x - r, x + r \rangle$$
+> Note:
+> $$\cap_{n \in \mathbb{N}} \langle -\frac{1}{n}, \frac{1}{n} \rangle = \{ 0\}$$
+> ($\cap_{n \in \mathbb{N}}$ is an infinite intersection of all numbers (in $\mathbb{N}$))
+> But the reason why this is not $= \{ 0, \varepsilon \}$ is a finite amount of intersections
+> $\varepsilon \notin \langle -\frac{1}{n}, \frac{1}{n} \rangle$ for $n \gt \frac{1}{\varepsilon}$
+

content/Definitions/Topological Spaces/Tychonoff Theorem.md

@@ -0,0 +1,3 @@
+Can be abbreviated as "THM".
+# Definition
+The product of compact spaces is compact in the [[Product Topology|product topology]]