---
title: Introduction to Topology
tags: Course
description:
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# Introduction to Topology
## Fundamental problem
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A topological space is a set $X$ together with a collection $\mathcal{T}$ of subsets of $X$ called "open" sets such that:
1. the intersection of two open sets is open;
2. the union of any collection of open sets is open; and
3. the empty set $\varnothing$ and whole space $X$ are open.

Additionally, a subset $C \subset X$ is called "closed" if its complement $X\setminus C$ is open.
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For example, given a set $A=\{1,2,3\}$
1. discrete topology: $\mathcal{T}=\mathcal{P}(A)$
2. trivial topology: $\mathcal{T}=\{\phi,A\}$
3. $\mathcal{T}=\{\{1\}, \{1,2\}, A, \phi\}$

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If $X$ is a topological space and $x \in X$ then a set $N$ is called a *neighborhood* of $x$ in $X$ if there is an open set $U \subset N$ with $x \in U$.
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If $X$ is a topological space, $\left(x_i\right)_{i=1}^{\infty}$ is a sequence of points in $X$, and $x \in X$, we say that the sequence *converges* to $x$, if for every neighborhood $U$ of $x$ there exists $N \in \mathbb{N}$ such that $x_i \in U$ for all $i \geq N$.
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### Continuous
The following definition is a local property.
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If $X$ and $Y$ are topological spaces and $f: X \rightarrow Y$ is a function, then $f$ is said to be continuous if $f^{-1}(U)$ is open for each open set $U \subset Y$. A map is a continuous function.
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The following definition is a global property.
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A function $f: X \rightarrow Y$ between topological spaces is called a *homeomorphism* if $f^{-1}: Y \rightarrow X$ exists (i.e., $f$ is one-one and onto) and both $f$ and $f^{-1}$ are continuous. The notation $X \approx Y$ means that $X$ is *homeomorphic* to $Y$.
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**Theorem.** Let $\mathcal{T}_1$ and $\mathcal{T}_2$ be topologies on the same set $X$. Show that the identity map of $X$ is continuous as a map from $\left(X, \mathcal{T}_1\right)$ to $\left(X, \mathcal{T}_2\right)$ if and only if $\mathcal{T}_1$ is finer than $\mathcal{T}_2$, and is a homeomorphism if and only if $\mathcal{T}_1=\mathcal{T}_2$.
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For example,
- Let $(M, d)$ be a metric space, let $c$ be a positive real number, and define a new metric $d^{\prime}$ on $M$ by $d^{\prime}(x, y)=c \cdot d(x, y)$. Then, $d$ and $d^{\prime}$ generate the same topology on $M$.
- Let $X$ be any set, and let $d$ be the discrete metric on $X$. Then, $d$ generates the discrete topology.

## Enugh?
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1. Suppose $(X,\mathcal{T})$ a topology space where $X=\{a,b\}$ and $\mathcal{T}=\{\{a\},X,\phi\}$. Then, $\{a\}$ is not closed.
2. Suppose $(X,\mathcal{T})$ a topology space where $X=\{1,2,3\}$ and $\mathcal{T}=\{\{1\}, \{1,2\}, X, \phi\}$. Then, $x_n=2\to 3$.
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Not to structures like a **metric**, but to conditions describable in terms of the topology itself.
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The separation axioms:
$\left(\mathrm{T}_0\right)$ A topological space $X$ is called a $\mathrm{T}_0$-space if for any two points $x \neq y$ there is an open set containing one of them but not the other.
$\left(\mathrm{T}_1\right)$ A topological space $X$ is called a $\mathrm{T}_1$-space if for any two points $x \neq y$ there is an open set containing $x$ but not $y$ and another open set containing $y$ but not $x$.
$\left(\mathrm{T}_2\right)$ A topological space $X$ is called a $\mathrm{T}_2$-space or ***Hausdorff*** if for any two points $x \neq y$ there are disjoint open sets $U$ and $V$ with $x \in U$ and $y \in V$.
$\left(\mathrm{T}_3\right)$ A $\mathrm{T}_1$-space $X$ is called a $\mathrm{T}_3$-space or *regular* if for any point $x$ and closed set $F$ not containing $x$ there are disjoint open sets $U$ and $V$ with $x \in U$ and $F \subset V$.
$\left(\mathrm{T}_4\right)$ A $\mathrm{T}_1$-space $X$ is called a $\mathrm{T}_4$-space or *normal* if for any two disjoint closed sets $F$ and $G$ there are disjoint open sets $U$ and $V$ with $F \subset U$ and $G \subset V$.
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For example,
- **Every metric space is Hausdorff**: if $p_1$ and $p_2$ are distinct, let $r=d\left(p_1, p_2\right)$; then the open balls of radius $r / 2$ around $p_1$ and $p_2$ are disjoint by the triangle inequality.

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**Theorem.** Let $X$ be a Hausdorff space.
(a) Every finite subset of $X$ is closed.
(b) If a sequence $\left(p_i\right)$ in $X$ converges to a limit $p \in X$, the limit is unique.
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## Too many?
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Let $\left\{X_\alpha\right\}_{\alpha \in J}$ be an indexed family of topological spaces. Let us take as a basis for a topology on the product space
\begin{equation*}
\prod_{\alpha \in J} X_\alpha
\end{equation*}
the collection of all sets of the form
\begin{equation*}
\prod_{\alpha \in J} U_\alpha,
\end{equation*}
where $U_\alpha$ is open in $X_\alpha$, for each $\alpha \in J$. The topology generated by this basis is called the *box topology*.
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Let $(\mathbb{R}^\omega,\mathcal{T})$ be topology space with box topology. Given any sequence $\{a_i=(a_{i1},a_{i2},\cdots):i=1,2,\cdots\}$, exist a neighborhood $U=(-a_{11},a_{11})\times (-a_{22},a_{22})\times\cdots$ such that $a_i$ cannot converge to $a$.
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### Basis
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Let $X$ be a topological space. A collection $\mathscr{B}$ of subsets of $X$ is called a basis for the topology of $X$ (plural: bases) if the following two conditions hold:
(i) Every element of $\mathscr{B}$ is an open subset of $X$.
(ii) Every open subset of $X$ is the union of some collection of elements of $\mathcal{B}$.
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For example,
- Let $M$ be a metric space. Every open ball in $M$ is an open subset, and every open subset is a union of open balls. Thus the collection of all open balls in $M$ is a basis for the metric topology.
- If $X$ is any set with the discrete topology, the collection of all singleton subsets of $X$ is a basis for its topology.

### Countablility
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The countability axioms.
- A topological space is said to be *first countable* if each point has a countable neighborhood basis.
- Definition. A topological space is said to be *second countable* if its topology has a countable basis.
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For example,
- If $X$ is a metric space and $p \in X$, the set of balls $B_r(p)$ with rational radii is easily seen to be a neighborhood basis at $p$, so every metric space is first countable.
- Every Euclidean space is second countable, because as Exercise 2.42(b) shows, it has a countable basis.

The properties of first countable
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**Lemma (Sequence Lemma).** Suppose $X$ is a first countable space, $A$ is any subset of $X$, and $x$ is any point of $X$.
(i) $x \in \bar{A}$ if and only if $x$ is a limit of a sequence of points in $A$.
(ii) $A$ is closed in $X$ if and only if $A$ contains every limit of every convergent sequence of points in $A$.
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The properties of second countable
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**Theorem.** Suppose $X$ is a second countable space.
(i) $X$ is first countable.
(ii) **(separable space)** $X$ contains a countable dense subset.
(iii) **(Lindelof space)** Every open cover of $X$ has a countable subcover.
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:::spoiler Click here to see exercise in [Rudin]
- 2.22 A metric space is called separable if it contains a countable dense subset. Show that $R^k$ is separable.
- 2.23 Prove that every separable metric space has a countable base.
- 2.24 Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is separable.
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For example, if $\mathcal{B}^{\prime}$ is the collection of all half open intervals in the real line,
\begin{equation*}
[a, b)=\{x \mid a \leq x<b\},
\end{equation*}
the topology generated by $\mathcal{B}^{\prime}$ is called the *lower limit topology (Sorgenfre line)* on $\mathbb{R}$. Denote it as $\mathbb{R}_\ell$.

- $\mathbb{R}_\ell$ is first-countable and separable, but not second-countable.
- Metric space is first countable, but might not be second countble and separable.

## Back to Metrization
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If $X$ is a topological space, $X$ is said to be metrizable if there exists a metric $d$ on the set $X$ that induces the topology of $X$.
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$\mathbb{R}_\ell$ is Hausdorff space but not metrizable. Exercise 2.23 in [Rudin] give a intuition that separable metric spaces are second-countable.
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When?
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**Theorem.** **(The Urysohn Metrization Theorem)** Every second countable regular space is metrizable.
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## Manifold
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A topological space $M$ is said to be *locally Euclidean* of dimension $\boldsymbol{n}$ if every point of $M$ has a neighborhood in $M$ that is homeomorphic to an open subset of $\mathbb{R}^n$.
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An n-dimensional topological manifold is a second countable Hausdorff space that is locally Euclidean of dimension $n$.
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[Line with two origins](https://topospaces.subwiki.org/wiki/Line_with_two_origins) locally Euclidean but non-Hausdorff.
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## To be continued..
"seperable" is important concept in function space!!

## References
1. J. Lee, *Introduction to Topological Manifolds*
2. G. Bredon, *Topology and Geometry*
3. J. Munkres, *Topology*