# Topological Spaces

## Elementary Concepts

Definition (Topology, Topological Space)

A

topologyon a set $X$ is a collection $T$ of subsets of $X$ such that

(T1)$∅$ and $X$ belong to $T$,

(T2)the union of any subcollection of $T$ belongs to $T$,

(T3)the intersection of any finite subcollection $T$ belongs to $T$.

Atopological spaceis a pair $(X,T)$ consisting of a set $X$ and a topology $T$ 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 $(X,T)$ is a topological space, the elements of $X$ are called *points* and the elements of $T$ are called the *open sets*.

Example On every set $X$ we have the *trivial* (or *indiscrete*) *topology* ${∅,X}$ and the *discrete topology* $P(X)$. 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 $T$ and $T_{′}$ are topologies on a set $X$. When $T⊂T_{′}$, we say that $T$ is

coarserorsmallerorweakerthan $T_{′}$, and that $T_{′}$ isfinerorlargerorstrongerthan $T$. If the inclusion is proper, then we saystrictly coarserand so on. If either $T⊂T_{′}$ or $T⊃T_{′}$ holds, then the topologies are said to becomparable.

Proposition (Intersection of Topologies)

If ${T_{α}}$ is a family of topologies on a set $X$, then $⋂_{α}T_{α}$ is a topology on $X$.

Definition (Generated Topology)

Suppose $A$ is a collection of subsets of a set $X$. The

topology generated by $A$is the intersection of all topologies on $X$ containing $A$.

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

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

## Bases and Subbases

Definition (Basis for a Topology)

A

basis for a topologyon a set $X$ is a collection $B$ of subsets of $X$ such that for every point $x∈X$

- there exists $B∈B$ such that $x∈B$,
- if $x∈B_{1}∩B_{2}$ for $B_{1},B_{2}∈B$, then there exists a $B_{3}∈B$ such that $x∈B_{3}⊂B_{1}∩B_{2}$.

Theorem (Topology Generated by a Basis)

If $X$ is set and $B$ is a basis for a topology on $X$, then the topology generated by $B$ equals

- the collection of all subsets $S⊂X$ with the property that for every $x∈S$ there exists a basis element $B∈B$ such that $x∈B$ and $B⊂S$;
- the collection of all arbitrary unions of elements of $B$.

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

Example If $X$ is a set, then the collection of singletons ${x}$, $x∈X$, 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 topologyon a set $X$ is a collection $S$ of subsets of $X$ such that for every point $x∈X$ there exists a $S∈S$ such that $x∈S$.

Theorem (Topology Generated by a Subbasis)

If $X$ is set and $S$ is a subbasis for a topology on $X$, then the topology generated by $S$ equals

- the collection of all arbitrary unions of finite intersections of elements of $S$.

## Open and Closed Sets

Definition (Open Set, Closed Set)

Suppose $(X,T)$ is a topological space. A subset $S$ of $X$ is called

openwith respect to $T$ when it belongs to $T$, and it is calledclosedwith respect to $T$ when its complement $X∖S$ belongs to $T$.

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

Let $C$ be the collection of closed subsets of a topological space. Then

- $X$ and $∅$ belong to $C$,
- the intersection of any subcollection of $C$ belongs to $C$,
- the union of any finite subcollection $C$ belongs to $C$.