Quartz sync: May 26, 2025, 10:16 AM

content/Definitions/Topological Spaces/Topological Space.md

@@ -1,4 +1,6 @@
 # Definition
+A [[Topological Space]] $X$ is a set with a [[Topology]] $\tau$.
+
 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 $A \cap B = \emptyset$.

content/Exam Preparation.md

@@ -5,10 +5,10 @@ The following list is meant to provide a starting point for the type of question
 ## Topology and Measure Theory
 Here are 20 relevant exam questions in the topology and measure theory part of the course:
 
-1. What distinguishes the real numbers from the rational ones?
-2. What is an equivalence relation?
-3. What is a topological space? Examples?
-4. What is the ball topology on a metric space?
+1. What distinguishes the real numbers from the rational ones? [[Question 1|Answer]]
+2. What is an equivalence relation? [[Question 2|Answer]]
+3. What is a topological space? Examples? [[Question 3|Answer]]
+4. What is the ball topology on a metric space? [[Question 4|Answer]]
 5. What is the topology on a Banach space?
 6. What is a compact set?
 7. State the Heine-Borel theorem. Proof?
@@ -21,10 +21,11 @@ Here are 20 relevant exam questions in the topology and measure theory part of t
 14. Define the Lebesgue integral of a extended non-negative measurable function.
 15. State Lebesgue's monotone convergence theorem.
 16. Define $L^p$-spaces, and point out their crucial property.
-17. (Not relevant: State the Riesz representation theorem.
+### Not relevant
+17. State the Riesz representation theorem.
 18. What is the Lebesgue measure on $\mathbb{R}^n$ ?
 19. What is a complex measure?
-20. State the Lebesgue-Radon-Nikodym theorem.)
+20. State the Lebesgue-Radon-Nikodym theorem.
 ## Complex Analysis
 - Similarities and differences between $\mathbb{C}$ and $\mathbb{R}^2$.
 - Holomorphic functions and their properties.

content/Lectures/Lecture 1 - 1.1 Sets and Numbers.md

@@ -74,7 +74,7 @@ These two elements $x$ and $y$ can only relate if they are the same.
 ### Equivalence Relation
 An **equivalence relation** $0 \sim X$ is a relation on $\sim$ on $X$ such that:
 1. $x \sim x$
-2. $x \sim y \implies y \sim x$
+2. $x \sim y \iff y \sim x$
 3. $(x \sim y) \cap (y \sim z) \implies x \sim z$
 For all $x,y,z \in X$.
 

content/Revision/Real Analysis/Question 1.md

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+# Question
+What distinguishes the real numbers from the rational ones?
+# Answer
+The main way to distinguish these is by knowing how they are constructed. And we get a set of these numbers based on how they are constructed.
+But the basic structure of what real numbers encompass is: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$. and $\mathbb{1} \subset \mathbb{R}$.
+## Rational Numbers
+Rational numbers are numbers that you can express in a fraction.
+For example $\left\{  1 \equiv \frac{1}{1}, 0.5 \equiv \frac{1}{2}, 2 \equiv \frac{2}{1}  \right\}$.
+## Irrational Numbers
+They are numbers that cannot be expressed as a fraction. These are numbers such as pi, which as of now has only been calculated to 300 trillion digits.

content/Revision/Real Analysis/Question 2.md

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+# Question
+What is an equivalence relation?
+# Answer
+An **equivalence relation** $0 \sim X$ is a relation on $\sim$ on $X$ such that:
+1. $x \sim x$ (reflexive)
+2. $x \sim y \iff y \sim x$ (symmetry)
+3. $(x \sim y) \cap (y \sim z) \implies x \sim z$ (transitivity)
+For all $x,y,z \in X$.
+
+Taken from [[Lecture 1 - 1.1 Sets and Numbers#Equivalence Relation]]

content/Revision/Real Analysis/Question 3.md

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+# Question
+What is a topological space? Examples?
+# Answer
+A [[Topological Space]] $X$ is a set with a [[Topology]] $\tau$.
+
+A topology has $X$ and $\emptyset$.
+Any union ($\cup$) of $\tau$ will be in $\tau$
+Any **finite** intersection ($\cap$) of sets from $\tau$ will be in $\tau$.
+## Example
+$X$ is Hausdorff if for $x \neq y\; \exists \, \text{neighbourhoods} \, A, B$ of $x$ and $y$ respectively, such that $A \cap B = \emptyset$.
+
+Basically, sets do not intersect on the topological space, making everything an open set.

content/Revision/Real Analysis/Question 4.md

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+# Question
+What is the ball topology on a metric space?
+# Answer