## Topological spaces

A topological space is a set $X$ with a specification of the open subsets of $X$ where it is required that

(a)   $\varnothing$ is open and $X$ is open,
(b)   Unions of open sets are open,
(c)   Finite intersections of open sets are open.
In other words, a topology on $X$ is a set $𝒯$ of subsets of $X$ such that
(a)   $\varnothing \in 𝒯$ and $X\in 𝒯$,
(b)   If $𝒮\subseteq 𝒯$ then $\left(\bigcup _{U\in 𝒮}U\right)\in 𝒯$,
(c)   If $n\in {ℤ}_{\ge 0}$ and ${U}_{1},{U}_{2},\dots ,{U}_{n}\in 𝒯$ then ${U}_{1}\cap {U}_{2}\cap \dots \cap {U}_{n}\in 𝒯.$

A topological space is a set $X$ with a topology $𝒯$ on $X$.

Let $X$ be a set and let $𝒯$ be a topology on $X$.

• An open set is a set in $𝒯$.
• A closed set is a subset $E$ of $X$ such that the complement ${E}^{c}$ of $E$ is open.
• A connected set is a subset $E\subseteq X$ such that there do not exist open sets $A$ and $B$ with $A∩E≠∅, B∩E≠∅, A∪B⊇E, and ( A∩B) ∩E=∅.$
• A compact set is a subset $E\subseteq X$ such that every open cover of $E$ contains a finite subcover.
More precisely, a compact set is a subset $E\subseteq X$ such that
if $𝒮\subseteq 𝒯$ and $\bigcup _{U\in 𝒮}U\supseteq E$
then there exists $n\in {ℤ}_{>0}$ and ${U}_{1},\dots ,{U}_{n}\in 𝒮$ such that $\left({U}_{1}\cup {U}_{2}\cup \dots \cup {U}_{n}\right)\supseteq E.$

HW: Let $X$ be a topological space. Let $𝒞$ be the collection of closed sets of $X\text{.}$ Show that $𝒞$ satisfies:

 (Ca) If $𝒮\subseteq 𝒞$ then $\bigcap _{C\in 𝒮}C\in 𝒞\text{.}$ (Cb) If $\ell \in {ℤ}_{>0}$ and ${C}_{1},{C}_{2},\dots ,{C}_{\ell }\in 𝒞$ then ${C}_{1}\cup \cdots \cup {C}_{\ell }\in 𝒞\text{.}$

HW: Let $X$ be a set and let $𝒞$ be a collection of subsets of $X$ which satisfies (Ca) and (Cb).

 (a) Show that $𝒯={A⊆X | Ac∈𝒞}$ is a topology on $X$ such that $𝒞$ is the set of closed sets in $X\text{.}$ (b) Show that $𝒯$ is the unique topology on $X$ such that $𝒞$ is the set of closed sets in $X\text{.}$

## Continuous Functions

Continuous functions are for comparing topological spaces.

Let $X$ and $Y$ be topological spaces.

• A function $f:X\to Y$ is continuous if it satisfies the condition  if $V$ is an open subset of $Y$     then     ${f}^{-1}\left(V\right)$ is an open subset of $X$.
• Let $X$ and $Y$ be topological spaces. An isomorphism, or homeomorphism, is a continuous function $f:X\to Y$ such that the inverse function ${f}^{-1}:Y\to X$ exists and is continuous.
• A subspace of $X$ is a subset $E$ of $X$ with the topology given by making the open sets be the sets $ι-1 (V) such that V is an open subset of X,$ where $\iota :E\to X$ is the inclusion.

Let $f:X\to Y$ be a continuous function and let $E\subseteq X$.

(a)   If $E$ is connected then $f\left(E\right)$ is connected.
(b)   If $E$ is compact then $f\left(E\right)$ is compact.

 Proof of (a): Proof by contradiction. Assume $f\left(E\right)$ is not connected. Let $A$ and $B$ be open in $f\left(E\right)$ such that $A\ne \varnothing$ and $B\ne \varnothing$ and $A\cup B\supseteq f\left(E\right)$ and $A\cap B=\varnothing$. Then let $C={f}^{-1}\left(A\right)$ and $D={f}^{-1}\left(B\right)$. Then $C\cup D={f}^{-1}\left(A\right)\cup {f}^{-1}\left(B\right)={f}^{-1}\left(A\cup B\right)\supseteq {f}^{-1}\left(f\left(E\right)\right)\supseteq E$, and $C\cap D={f}^{-1}\left(A\right)\cap {f}^{-1}\left(B\right)={f}^{-1}\left(A\cap B\right)={f}^{-1}\left(\varnothing \right)=\varnothing$. Now $C\ne \varnothing$ since $A\ne \varnothing$ and $A\subseteq f\left(E\right)$, and $D\ne \varnothing$ since $B\ne \varnothing$ and $B\subseteq f\left(E\right)$. So $E$ is not connected. This is a contradiction. So $f\left(E\right)$ is connected. $\square$

 Proof of (b): Assume $f:X\to Y$ is continuous and $E$ is compact. To show: If $𝒮$ is an open cover of $f\left(E\right)$ then it has a finite subcover. To show: If $𝒴$ is the topology on $Y$, $𝒮\subseteq 𝒴$, and $\left(\bigcup _{V\in 𝒮}V\right)\supseteq f\left(E\right)$ then there exists $n\in {ℤ}_{>0}$ and ${V}_{1},\dots ,{V}_{n}\in 𝒮$ such that ${V}_{1}\cup \dots \cup {V}_{n}\supseteq f\left(E\right)$. Assume $𝒴$ is the topology on $Y$ and $𝒮\subseteq 𝒴$ and $\left(\bigcup _{V\in 𝒮}V\right)\supseteq f\left(E\right)$. Then $\left(\bigcup _{V\in 𝒮}{f}^{-1}\left(V\right)\right)\supseteq E$. Let $𝒯=\left\{{f}^{-1}\left(V\right)\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}V\in 𝒮\right\}$. Since $f$ is continuous, ${f}^{-1}\left(V\right)$ is open. Thus $𝒯\subseteq 𝒳$ where $𝒳$ is the topology of $X$, and also $\left(\bigcup _{{f}^{-1}\left(V\right)\in 𝒯}{f}^{-1}\left(V\right)\right)\supseteq E$. Since $E$ is compact there exists $n\in {ℤ}_{>0}$ and ${f}^{-1}\left({V}_{1}\right),\dots ,{f}^{-1}\left({V}_{n}\right)\in 𝒯$ such that $\left({f}^{-1}\left({V}_{1}\right)\cup \dots \cup {f}^{-1}\left({V}_{n}\right)\right)\supseteq E$. So $\left({V}_{1}\cup \dots \cup {V}_{n}\right)\supseteq f\left(E\right)$. Thus $𝒮$ contains a finite subcover of $f\left(E\right)$. $\square$

Let $Y$ be a topological space. Let $y\in Y$.

• A neighbourhood of $y$ is a subset $N\subseteq Y$ such that there exists an open set $U$ of $Y$ with $y\in U\subseteq N$.
• The neighbourhood filter of $y$ is  $𝒩\left(y\right)=\left\{\text{neighborhoods of}\phantom{\rule{0.5em}{0ex}}y\right\}$.

Let $X$ and $Y$ be topological spaces. Let $a\in X$.

• A function $f:X\to Y$ is continuous at $a$ if it satisfies the condition  if $V$ is a neighborhood of $f\left(a\right)$ in $Y$     then     ${f}^{-1}\left(V\right)$ is a neighborhood of $a$ in $X$.
Here ${f}^{-1}\left(V\right)=\left\{x\in X\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}f\left(x\right)\in V\right\}.$

HW: Let $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a function. Show that $f:X\to Y$ is continuous if and only if $fsatisfies: if a∈X then fis continuous at a.$

HW: Let $X$ be a topological space. For $x\in X$ let $𝒩\left(x\right)$ be the set of neighborhoods of $x\text{.}$ Show that the collections $𝒩\left(X\right)$ satisfy:

 $\left({\text{V}}_{\text{I}}\right)$ If $A\subseteq X$ and there exists $N\in 𝒩\left(x\right)$ such that $A\supseteq N$ then $A\in 𝒩\left(x\right)\text{.}$ $\left({\text{V}}_{\text{II}}\right)$ If $\ell \in {ℤ}_{>0}$ and ${N}_{1},{N}_{2},\dots ,{N}_{\ell }\in 𝒩\left(x\right)$ then ${N}_{1}\cap {N}_{2}\cap \dots \cap {N}_{\ell }\in 𝒩\left(x\right)\text{.}$ $\left({\text{V}}_{\text{III}}\right)$ If $N\in 𝒩\left(x\right)$ then $x\in N\text{.}$ $\left({\text{V}}_{\text{IV}}\right)$ If $N\in 𝒩\left(x\right)$ then there exists $W\in 𝒩\left(x\right)$ such that if $y\in W$ then $N\in 𝒩\left(y\right)\text{.}$

HW: Let $X$ be a set with a collection of subsets of $X,$ $𝒩\left(x\right)$ for each $x\in X,$ such that the $𝒩\left(x\right)$ satisfy $\left({\text{V}}_{\text{I}}\right),\left({\text{V}}_{\text{II}}\right),\left({\text{V}}_{\text{III}}\right),\left({\text{V}}_{\text{IV}}\right)\text{.}$ Let $𝒯= { A⊆X | if x∈A then A∈ 𝒩(x) } .$

 (a) Show that $𝒯$ is a topology on $X\text{.}$ (b) Show that the $𝒩\left(x\right),$ $x\in X,$ are the neighborhood filters for the topology $𝒯\text{.}$ (c) Show that $𝒯$ is the unique topology on $X$ such that $𝒩\left(x\right),$ $x\in X,$ are the neighborhood filters for $𝒯\text{.}$

Let $I$ be a set and let ${X}_{i},$ $i\in I,$ be topological spaces.

• The product space $\prod _{i\in I}{X}_{i}$ is the set $\prod _{i\in I}{X}_{i}$ with topology generated by the sets $pri(Ui) with Ui open in Xi,$ where ${\text{pr}}_{i}:\prod _{i\in I}{X}_{i}\to {X}_{i}$ is the projection (onto the ${i}^{\text{th}}$ coordinate).

HW: Show that the topology on $\prod _{i\in I}{X}_{i}$ is the weakest topology such that the projections ${\text{pr}}_{i}:\prod _{i\in I}{X}_{i}\to {X}_{i},$ for $i\in I,$ are continuous.

## Examples

(1) Let $X$ be a set. The discrete topology on $X$ is the topology such that every subset of $X$ is open.

(2) A metric space is a set $X$ with a function $d:X×X\to {ℝ}_{\ge 0}$ such that

(a)   If $x\in X$ then $d\left(x,x\right)=0$,
(b)   If $x,y\in X$ and $d\left(x,y\right)=0$, then $x=y$,
(c)   If $x,y,z\in X$ then $d\left(x,z\right)\le d\left(x,y\right)+d\left(y,z\right)$.

Let $X$ be a metric space. Let $x\in X$ and let $\epsilon \in {ℝ}_{>0}$. The ball of radius $\epsilon$ at $x$ is the set $Bε(x) ={p∈X | d(x,y) <ε}.$ Let $X$ be a metric space. The metric space topology on $X$ is the topology generated by the sets $Bε(x) , for x∈X and ε∈ ℝ>0.$

HW: Show that the metric space topology is a topology on $X$.

HW: Let $X=\left\{0,1\right\}$ and let $𝒯=\left\{\varnothing ,\left\{0\right\},X\right\}$. Show that $𝒯$ is a topology on $X$ and that there does not exist a metric $d:X×X\to {ℝ}_{\ge 0}$ on $X$ such that $𝒯$ is the metric space topology on $X$.

## Generating topologies

Let $X$ be a topological space. Let $x\in X\text{.}$

• A base of the topology on $X$ is a collection $ℬ$ of open sets of $X$ such that if $U$ is open in $X$ then there exists $𝒮\subseteq ℬ$ such that $U=\bigcup _{A\in ℬ}A\text{.}$
• A fundamental system of neighborhoods of $X$ is a set $𝒮$ of neighborhoods of $X$ such that if $N\in 𝒩\left(x\right)$ then there exists $W\in 𝒮$ such that $W\subseteq N\text{.}$

HW: Show that $ℬ$ is a base of the topology on $X$ if and only if $ℬ$ satisfies if $x\in X$ then $\left\{V\in ℬ | x\in V\right\}$ is a fundamental system of neighborhoods of $x\text{.}$

This is found in [Bou, Ch I §1.3 Proposition 3].

HW: Show that the topology on $\prod _{i\in I}{X}_{i}$ has base ${ ∏i∈IAi | Ai is open in Xi, Ai=Xi for all but a finite number of i∈I } .$

HW: Let $x\in \prod _{i\in I}{X}_{i}\text{.}$ Show that ${ ∏i∈IAi | Ai is open in Xi, Ai=Xi for all but a finite number of i∈I, x∈∏i∈IAi }$ is a fundamental system of neighborhoods of $x\text{.}$

## Homework

1. ${f}^{-1}\left(C\cup D\right)={f}^{-1}\left(C\right)\cup {f}^{-1}\left(D\right)$.
2. ${f}^{-1}\left(C\cap D\right)={f}^{-1}\left(C\right)\cap {f}^{-1}\left(D\right)$.
3. ${f}^{-1}\left(f\left(E\right)\right)\supseteq E$.
4. Give an example where ${f}^{-1}\left(f\left(E\right)\right)\ne E$.
5. If $E$ is compact then $E$ is closed.

## Notes and References

This material is found in almost every textbook on introductory topology or mathematical analysis. One comprehensive reference is Bourbaki -- see [Bou, Ch I § 1 nos. 1,2,4] for open sets, closed sets and neighborhoods, [Bou, Ch I § 3 no. 1] for subspaces, [Bou Ch I § 4 no. 1] for products, [Bou Ch I § 9 no. 1 Axiom C'''] and [Bou, Ch. I § 9 no. 4] for compact sets, [Bou, Ch. I § 11 nos. 1,2] for connected sets, and [Bou, Ch. I § 2 no. 1] for continuous functions.

The most important theorem in the theory of continuous functions is the following. This theorem will be proved in the section on Limits in Topological spaces.

Let $X$ and $Y$ be topological spaces and let $a\in X$. A function

 $f:X\to Y$ is continuous at $a$    if and only if    $\underset{x\to a}{lim}f\left(x\right)=f\left(a\right).$

## References

[Bou] N. Bourbaki, General Topology, Springer-Verlag, 1989. MR1726779.

[BR] W. Rudin, Principles of mathematical analysis, Third edition, International Series in Pure and Applied Mathematics, McGraw-Hill 1976. MR0385023.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.