---
title: Topological Spaces
parent: General Topology
nav_order: 1
---

# {{ page.title }}

## Elementary Concepts

{% definition Topology, Topological Space %}
A *topology* on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ such that \
**(T1)** $\varnothing$ and $X$ belong to $\mathcal{T}$, \
**(T2)** the union of any subcollection of $\mathcal{T}$ belongs to $\mathcal{T}$, \
**(T3)** the intersection of any finite subcollection $\mathcal{T}$ belongs to $\mathcal{T}$. \
A *topological space* is a pair $(X,\mathcal{T})$ consisting of
a set $X$ and a topology $\mathcal{T}$ on $X$.
{% enddefinition %}

If one follows the convention that
the union of the empty collection of subsets of $X$ is the empty subset of $X$,
and its intersection is all of $X$,
then **(T1)** is a consequence of **(T2)**, **(T3)**
and can be omitted.

If $(X,\mathcal{T})$ is a topological space,
the elements of $X$ are called *points*
and the elements of $\mathcal{T}$ are called the *open sets*.

{% example %}
On every set $X$ we have
the *trivial* (or *indiscrete*) *topology* $\braces{\varnothing,X}$ and
the *discrete topology* $\mathcal{P}(X)$.
These collections are in fact topologies on $X$.
{% endexample %}

{% example %}
If $X$ is any set,
then the collection of all subsets of $X$
whose complement is either finite or all of $X$
is a topology on $X$;
it is called the *finite complement topology*.
The *countable complement topology* is defined analogously.
{% endexample %}

{% definition Comparison of Topologies %}
Suppose $\mathcal{T}$ and $\mathcal{T}'$ are topologies on a set $X$.
When $\mathcal{T} \subset \mathcal{T}'$,
we say that $\mathcal{T}$ is *coarser* or *smaller* or *weaker* than $\mathcal{T}'$,
and that $\mathcal{T}'$ is *finer* or *larger* or *stronger* than $\mathcal{T}$.
If the inclusion is proper, then we say *strictly coarser* and so on.
If either $\mathcal{T} \subset \mathcal{T}'$ or $\mathcal{T} \supset \mathcal{T}'$ holds,
then the topologies are said to be *comparable*.
{% enddefinition %}

{% proposition Intersection of Topologies %}
If $\braces{\mathcal{T}_{\alpha}}$ is a family of topologies on a set $X$,
then $\bigcap_{\alpha} \mathcal{T}_{\alpha}$ is a topology on $X$.
{% endproposition %}

{% definition Generated Topology %}
Suppose $\mathcal{A}$ is a collection of subsets of a set $X$.
The *topology generated by $\mathcal{A}$* is
the intersection of all topologies on $X$ containing $\mathcal{A}$.
{% enddefinition %}

By the previous proposition, the generated topology is indeed a topology.

{% proposition %}
The topology generated by a collection $\mathcal{A}$ of subsets of a set $X$
is the smallest topology on $X$ containing $\mathcal{A}$.
{% endproposition %}

## Bases and Subbases

{% definition Basis for a Topology %}
A *basis for a topology* on a set $X$ is a collection $\mathcal{B}$ of subsets of $X$
such that for every point $x \in X$

- there exists $B \in \mathcal{B}$ such that $x \in B$,
- if $x \in B_1 \cap B_2$ for $B_1, B_2 \in \mathcal{B}$,
  then there exists a $B_3 \in \mathcal{B}$
  such that $x \in B_3 \subset B_1 \cap B_2$.
{% enddefinition %}

{% theorem Topology Generated by a Basis %}
If $X$ is set and $\mathcal{B}$ is a basis for a topology on $X$,
then the topology generated by $\mathcal{B}$ equals

- the collection of all subsets $S \subset X$ with the property
  that for every $x \in S$ there exists a basis element $B \in \mathcal{B}$
  such that $x \in B$ and $B \subset S$;
- the collection of all arbitrary unions of elements of $\mathcal{B}$.
{% endtheorem %}

Let $\mathcal{T}$ be a topology on a set $X$.
As one might expect,
a collection $\mathcal{B}$ of subsets of $X$
is said to be a *basis for the topology $\mathcal{T}$*,
if $\mathcal{B}$ is basis for a topology on $X$ and
the topology generated by $\mathcal{B}$ equals $\mathcal{T}$.

{% example %}
If $X$ is a set, then the collection of singletons $\braces{x}$, $x \in X$,
is a basis for the discrete topology on $X$.
{% endexample %}

{% example %}
If $(X,d)$ is a metric space,
then the collection of open balls is a basis for the metric topology on $X$.
{% endexample %}

{% definition Subbasis for a Topology %}
A *subbasis for a topology* on a set $X$ is a collection $\mathcal{S}$ of subsets of $X$
such that for every point $x \in X$ there exists a $S \in \mathcal{S}$ such that $x \in S$.
{% enddefinition %}

{% theorem Topology Generated by a Subbasis %}
If $X$ is set and $\mathcal{S}$ is a subbasis for a topology on $X$,
then the topology generated by $\mathcal{S}$ equals
- the collection of all arbitrary unions of finite intersections of elements of $\mathcal{S}$.
{% endtheorem %}

## Open and Closed Sets

{% definition Open Set, Closed Set %}
Suppose $(X,\mathcal{T})$ is a topological space.
A subset $S$ of $X$
is called *open* with respect to $\mathcal{T}$
when it belongs to $\mathcal{T}$,
and it is called *closed* with respect to $\mathcal{T}$
when its complement $X \setminus S$ belongs to $\mathcal{T}$.
{% enddefinition %}

A subset of a topological space is open
if and only if its complement is closed.

{% proposition %}
Let $\mathcal{C}$ be the collection of closed subsets of a topological space. Then
{: .mb-0 }
- $X$ and $\varnothing$ belong to $\mathcal{C}$,
- the intersection of any subcollection of $\mathcal{C}$ belongs to $\mathcal{C}$,
- the union of any finite subcollection $\mathcal{C}$ belongs to $\mathcal{C}$.
{% endproposition %}

## The Subspace Topology