# 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 $[x]=\{y\in X|y\sim x\}$.
