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}

Warning

A set is closed is not the same as its not open.

In all topologies over non-empty sets, X, \emptyset are both closed and open.

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.

Definition of product topology

Let (X,\mathcal{T}_X) and (Y,\mathcal{T}_Y) be topological spaces. Then the product topology on X\times Y is the topology generated by the basis

\mathcal{B}_{X\times Y}=\{U\times V, U\in \mathcal{T}_X, V\in \mathcal{T}_Y\}

or equivalently,

\mathcal{B}_{X\times Y}'=\{U\times V, U\in \mathcal{B}_X, V\in \mathcal{B}_Y\}

Product topology generated from open sets of X and Y is the same as product topology generated from their corresponding basis

Subspace topology

Definition of subspace topology

Let (X,\mathcal{T}) be a topological space and Y\subseteq X. Then the subspace topology on Y is the topology given by

\mathcal{T}_Y=\{U\cap Y|U\in \mathcal{T}\}

or equivalently, let \mathcal{B} be the basis for (X,\mathcal{T}). Then the subspace topology on Y is the topology generated by the basis

\mathcal{B}_Y=\{U\cap Y| U\in \mathcal{B}\}

Lemma of open sets in subspace topology

Let (X,\mathcal{T}) be a topological space and Y\subseteq X. Then if U\subseteq Y, U is open in (Y,\mathcal{T}_Y), then U is open in (X,\mathcal{T}).

This also holds for closed set in closed subspace topology

Interior and closure

Definition of interior

The interior of A is the largest open subset of A.

A^\circ=\bigcup_{U\subseteq A, U\text{ is open in }X} U

Definition of closure

The closure of A is the smallest closed superset of A.

\overline{A}=\bigcap_{U\supseteq A, U\text{ is closed in }X} U

Definition of neighborhood

A neighborhood of a point x\in X is an open set U\in \mathcal{T} such that x\in U.

Definition of limit points

A point x\in X is a limit point of A if every neighborhood of x contains a point in A-\{x\}.

We denote the set of all limits points of A by A'.

\overline{A}=A\cup A'

Sequences and continuous functions

Definition of convergence

Let X be a topological space. A sequence (x_n)_{n\in\mathbb{N}_+} in X converges to x\in X if for any neighborhood U of x, there exists N\in\mathbb{N}_+ such that \forall n\geq N, x_n\in U.

Definition of Hausdoorff space

A topological space (X,\mathcal{T}) is Hausdorff if for any two distinct points x,y\in X, there exist open neighborhoods U and V of x and y respectively such that U\cap V=\emptyset.

Uniqueness of convergence in Hausdorff spaces

In a Hausdorff space, if a sequence (x_n)_{n\in\mathbb{N}_+} converges to x\in X and y\in X, then x=y.

Closed singleton in Hausdorff spaces

In a Hausdorff space, if x\in X, then \{x\} is a closed set.

Definition of continuous function

Let (X,\mathcal{T}_X) and (Y,\mathcal{T}_Y) be topological spaces. A function f:X\to Y is continuous if for any open set U\subseteq Y, f^{-1}(U) is open in X.

Definition of point-wise continuity

Let (X,\mathcal{T}_X) and (Y,\mathcal{T}_Y) be topological spaces. A function f:X\to Y is point-wise continuous at x\in X if for every openset V\subseteq Y, f(x)\in V then there exists an open set U\subseteq X such that x\in U and f(U)\subseteq V.

Lemma of continuous functions

If f:X\to Y is point-wise continuous for all x\in X, then f is continuous.

Properties of continuous functions

If f:X\to Y is continuous, then

1. \forall A\subseteq Y, f^{-1}(A^c)=X\setminus f^{-1}(A) (complements maps to complements)
2. \forall A_\alpha\subseteq Y, \alpha\in I, f^{-1}(\bigcup_{\alpha\in I} A_\alpha)=\bigcup_{\alpha\in I} f^{-1}(A_\alpha)
3. \forall A_\alpha\subseteq Y, \alpha\in I, f^{-1}(\bigcap_{\alpha\in I} A_\alpha)=\bigcap_{\alpha\in I} f^{-1}(A_\alpha)
4. f^{-1}(U) is open in X for any open set U\subseteq Y.
5. f is continuous at x\in X.
6. f^{-1}(V) is closed in X for any closed set V\subseteq Y.
7. Assume \mathcal{B} is a basis for Y, then f^{-1}(\mathcal{B}) is open in X for any B\in \mathcal{B}.
8. \forall A\subseteq X, \overline{f(A)}=f(\overline{A})

Definition of homeomorphism

Let (X,\mathcal{T}_X) and (Y,\mathcal{T}_Y) be topological spaces. A function f:X\to Y is a homeomorphism if f is continuous, bijective and f^{-1}:Y\to X is continuous.

Ways to construct continuous functions

1. If f:X\to Y is constant function, f(x)=y_0 for all x\in X, then f is continuous. (constant functions are continuous)
2. If A is a subspace of X, f:A\to X is the inclusion map f(x)=x for all x\in A, then f is continuous. (inclusion maps are continuous)
3. If f:X\to Y is continuous, g:Y\to Z is continuous, then g\circ f:X\to Z is continuous. (composition of continuous functions is continuous)
4. If f:X\to Y is continuous, A is a subspace of X, then f|_A:X\to Y is continuous. (domain restriction is continuous)
5. If f:X\to Y is continuous, Z is a subspace of Y, then f:X\to Z, g(x)=f(x)\cap Z is continuous. If Y is a subspace of Z, then h:X\to Z, h(x)=f(x) is continuous (composition of f and inclusion map).
6. If f:X\to Y is continuous, X can be written as a union of open sets \{U_\alpha\}_{\alpha\in I}, then f|_{U_\alpha}:X\to Y is continuous.
7. If X=Z_1\cup Z_2, and Z_1,Z_2 are closed equipped with subspace topology, let g_1:Z_1\to Y and g_2:Z_2\to Y be continuous, and for all x\in Z_1\cap Z_2, g_1(x)=g_2(x), then f:X\to Y by f(x)\begin{cases}g_1(x), & x\in Z_1 \\ g_2(x), & x\in Z_2\end{cases} is continuous. (pasting lemma)
8. f:X\to Y is continuous, g:X\to Z is continuous if and only if H:X\to Y\times Z, where Y\times Z is equipped with the product topology, H(x)=(f(x),g(x)) is continuous. (proved in homework)

Metric spaces

Definition of metric

A metric on X is a function d:X\times X\to \mathbb{R} such that \forall x,y\in X,

1. d(x,x)=0
2. d(x,y)\geq 0
3. d(x,y)=d(y,x)
4. d(x,y)+d(y,z)\geq d(x,z)

Definition of metric ball

The metric ball B_r^{d}(x) is the set of all points y\in X such that d(x,y)\leq r.

Definition of metric topology

Let X be a metric space with metric d. Then X is equipped with the metric topology generated by the metric balls B_r^{d}(x) for r>0.

Definition of metrizable

A topological space (X,\mathcal{T}) is metrizable if it is the metric topology for some metric d on X.

Hausdorff axiom for metric spaces

Every metric space is Hausdorff (take metric balls B_r(x) and B_r(y), r=\frac{d(x,y)}{2}).

If a topology isn't Hausdorff, then it isn't metrizable.

Prove by triangle inequality and contradiction.

Common metrics in \mathbb{R}^n

Euclidean metric

d(x,y)=\sqrt{\sum_{i=1}^n (x_i-y_i)^2}

Square metric

\rho(x,y)=\max_{i=1}^n |x_i-y_i|

Manhattan metric

m(x,y)=\sum_{i=1}^n |x_i-y_i|

These metrics are equivalent.

Product topology and metric

If (X,d),(Y,d') are metric spaces, then X\times Y is metric space with metric d(x,y)=\max\{d(x_1,y_1),d(x_2,y_2)\}.

Uniform metric

Let \mathbb{R}^\omega be the set of all infinite sequences of real numbers. Then \overline{d(x,y)}=\sup_{i=1}^\omega \min\{1,|x_i-y_i|\}, the uniform metric on \mathbb{R}^\omega is a metric.

Metric space and converging sequences

Let X be a topological space, A\subseteq X, x_n\to x such that x_n\in A. Then x\in \overline{A}.

If X is a metric space, A\subseteq X, x\in \overline{A}, then there exists converging sequence x_n\to x such that x_n\in A.

First countability axiom

A topological space (X,\mathcal{T}) satisfies the first countability axiom if any point x\in X, there is a sequence of open neighborhoods of x, \{V_n\}_{n=1}^\infty such that any open neighborhood U of x contains one of V_n.

Apply the theorem above, we have if (X,\mathcal{T}) satisfies the first countability axiom, then every convergent sequence converges to a point in the closure of the sequence.

Metric defined for functions

Definition for bounded metric space

A metric space (Y,d) is bounded if there is M\in \mathbb{R}^{\geq 0} such that for all y_1,y_2\in Y, d(y_1,y_2)\leq M.

Definition for metric defined for functions

Let X be a topological space and Y be a bounded metric space, then the set of all maps, denoted by \operatorname{Map}(X,Y), f:X\to Y\in \operatorname{Map}(X,Y) is a metric space with metric \rho(f,g)=\sup_{x\in X} d(f(x),g(x)).

Space of continuous map is closed

Let (\operatorname{Map}(X,Y),\rho) be a metric space defined above, then every continuous map is a limit point of some sequence of continuous maps.

Z=\{f\in \operatorname{Map}(X,Y)|f\text{ is continuous}\}

Z is closed in (\operatorname{Map}(X,Y),\rho).

Quotient space

Quotient map

Let X be a topological space and X^* is a set. q:X\to X^* is a surjective map. Then q is a quotient map.

Quotient topology

Let (X,\mathcal{T}) be a topological space and X^* be a set, q:X\to X^* is a surjective map. Then

\mathcal{T}^* \coloneqq \{U\subseteq X^*\mid q^{-1}(U)\in \mathcal{T}\}

is a topology on X^* called quotient topology.

(X^*,\mathcal{T}^*) is called the quotient space of X by q.

Equivalent classes

\sim is a subset of X\times X with the following properties:

1. x\sim x for all x\in X.
2. If (x,y)\in \sim, then (y,x)\in \sim.
3. If (x,y)\in \sim and (y,z)\in \sim, then (x,z)\in \sim.

The equivalence classes of x\in X is denoted by [x]=\{y\in X|y\sim x\}.