NoteNextra-origin/content/Math4201/Exam_reviews/Math4201_E1.md

Math 4201 Exam 1 review

Note

This is a review for definitions we covered in the classes. It may serve as a cheat sheet for the exam if you are allowed to use it.

The exam will have 5 problems, roughly covering the following types of questions:

• Define concepts from class (e.g. what is the definition of the interior of a set?)
• Give an example of a space/map which satisfies/does not satisfy a certain property (e.g. give an example of a map that is not continuous.)
• Proofs from the lectures
• Homework problems
• A new problem at the same level of difficulty as homework problems

Topological space

Basic definitions

Definition for topological space

A topological space is a pair of set X and a collection of subsets of X, denoted by \mathcal{T} (imitates the set of "open sets" in X), satisfying the following axioms:

1. \emptyset \in \mathcal{T} and X \in \mathcal{T}
2. \mathcal{T} is closed with respect to arbitrary unions. This means, for any collection of open sets \{U_\alpha\}_{\alpha \in I}, we have \bigcup_{\alpha \in I} U_\alpha \in \mathcal{T}
3. \mathcal{T} is closed with respect to finite intersections. This means, for any finite collection of open sets \{U_1, U_2, \ldots, U_n\}, we have \bigcap_{i=1}^n U_i \in \mathcal{T}

Definition of open set

U\subseteq X is an open set if U\in \mathcal{T}

Definition of closed set

Z\subseteq X is a closed set if X\setminus Z\in \mathcal{T}

Basis

Definition of topological basis

For a set X, a topology basis, denoted by \mathcal{B}, is a collection of subsets of X, such that the following properties are satisfied:

1. For any x \in X, there exists a B \in \mathcal{B} such that x \in B (basis covers the whole space)
2. If B_1, B_2 \in \mathcal{B} and x \in B_1 \cap B_2, then there exists a B_3 \in \mathcal{B} such that x \in B_3 \subseteq B_1 \cap B_2 (every non-empty intersection of basis elements are also covered by a basis element)

Definition of topology generated by basis

Let \mathcal{B} be a basis for a topology on a set X. Then the topology generated by \mathcal{B} is defined by the set as follows:

\mathcal{T}_{\mathcal{B}} \coloneqq \{ U \subseteq X \mid \forall x\in U, \exists B\in \mathcal{B} \text{ such that } x\in B\subseteq U \}

This is basically a closure of \mathcal{B} under arbitrary unions and finite intersections

Lemma of topology generated by basis

U\in \mathcal{T}_{\mathcal{B}}\iff \exists \{B_\alpha\}_{\alpha \in I}\subseteq \mathcal{B} such that U=\bigcup_{\alpha \in I} B_\alpha

Definition of basis generated from a topology

Let (X, \mathcal{T}) be a topological space. Then the basis generated from a topology is \mathcal{C}\subseteq \mathcal{B} such that \forall U\in \mathcal{T}, \forall x\in U, \exists B\in \mathcal{C} such that x\in B\subseteq U.

Definition of subbasis of topology

A subbasis of a topology is a collection \mathcal{S}\subseteq \mathcal{T} such that \bigcup_{U\in \mathcal{S}} U=X.

Definition of topology generated by subbasis

Let \mathcal{S}\subseteq \mathcal{T} be a subbasis of a topology on X, then the basis generated by such subbasis is the closure of finite intersection of \mathcal{S}

\mathcal{B}_{\mathcal{S}} \coloneqq \{B\mid B\text{ is the intersection of a finite number of elements of }\mathcal{S}\}

Then the topology generated by \mathcal{B}_{\mathcal{S}} is the subbasis topology denoted by \mathcal{T}_{\mathcal{S}}.

Note that all open set with respect to \mathcal{T}_{\mathcal{S}} can be written as a union of finitely intersections of elements of \mathcal{S}

Comparing topologies

Definition of finer and coarser topology

Let (X,\mathcal{T}) and (X,\mathcal{T}') be topological spaces. Then \mathcal{T} is finer than \mathcal{T}' if \mathcal{T}'\subseteq \mathcal{T}. \mathcal{T} is coarser than \mathcal{T}' if \mathcal{T}\subseteq \mathcal{T}'.

Lemma of comparing basis

Let (X,\mathcal{T}) and (X,\mathcal{T}') be topological spaces with basis \mathcal{B} and \mathcal{B}'. Then \mathcal{T} is finer than \mathcal{T}' if and only if for any x\in X, x\in B', B'\in \mathcal{B}', there exists B\in \mathcal{B}, such that x\in B and x\in B\subseteq B'.

Product space

Definition of cartesian product

Let X,Y be sets. The cartesian product of X and Y is the set of all ordered pairs (x,y) where x\in X and y\in Y, denoted by X\times Y.