Topological Spaces

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 $X$.

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 $\{\varnothing ,X\}$ and the discrete topology $\mathcal{P}(X)$. These collections are in fact topologies on $X$.

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.

Definition (Comparison of Topologies)

Suppose $\mathcal{T}$ and ${\mathcal{T}}^{\prime }$ are topologies on a set $X$. When $\mathcal{T}\subset {\mathcal{T}}^{\prime }$, we say that $\mathcal{T}$ is coarser or smaller or weaker than ${\mathcal{T}}^{\prime }$, and that ${\mathcal{T}}^{\prime }$ is finer or larger or stronger than $\mathcal{T}$. If the inclusion is proper, then we say strictly coarser and so on. If either $\mathcal{T}\subset {\mathcal{T}}^{\prime }$ or $\mathcal{T}\supset {\mathcal{T}}^{\prime }$ holds, then the topologies are said to be comparable.

Proposition (Intersection of Topologies)

If $\{{\mathcal{T}}_{\alpha }\}$ is a family of topologies on a set $X$, then ${\bigcap }_{\alpha }{\mathcal{T}}_{\alpha }$ is a topology on $X$.

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 $\mathcal{A}$.

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

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

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 $x\in B_3\subset B_1\cap B_2$.

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 $\mathcal{B}$.

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 $\{x\}$, $x\in X$, is a basis for the discrete topology on $X$.

Example   If $(X,d)$ is a metric space, then the collection of open balls is a basis for the metric topology on $X$.

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 $x\in S$.

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 $\mathcal{S}$.

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 $\mathcal{T}$.

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

Let $\mathcal{C}$ be the collection of closed subsets of a topological space. Then

• $X$ 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 $\mathcal{C}$.

The Subspace Topology