2.1Topological Spaces

Updated 31 May 2026Chapter (PDF)

[2.1.1]Definition(Topology, Topological Space)#

A topology on a set $X$ is a collection $\Tau$ of subsets of $X$ with the following properties.

$\emptyset$ and $X$ are elements of $\Tau$ .
The union of the elements of any subcollection of $\Tau$ is an element of $\Tau$ .
The intersection of the elements of any finite subcollection of $\Tau$ is an element of $\Tau$ .

A set $X$ endowed with some topology $\Tau$ is said to be a topological space.

[2.1.2]Definition(Open Set)#

If $(X, \Tau)$ is some topological space, we say $U \subseteq X$ is open if $U \in \Tau$ .

[2.1.3]Remark(Reformulation of Topological Space)#

With this, we may reformulate a topological space to be a set $X$ together with some collection of subsets, called open sets, such that $\emptyset$ and $X$ are open, as well as the arbitrary union of open sets being open, and the finite intersection of open sets being open.

[2.1.4]Example(Discrete, Indiscrete/Trivial Topology)#

For any set $X$ , the discrete topology is said to be the collection of every single subset of $X$ . The indiscrete/trivial topology is said to be the collection containing only $\emptyset$ and $X$ .

[2.1.5]Example(Finite Complement Topology)#

Fix a set $X$ . Define a topology $\Tau_f$ such that $U \subseteq X$ is in the topology if $X \setminus U$ is either finite or $X$ . Immediately we see $\emptyset$ and $X$ are elements of the topology, so now we must show the latter two axioms hold true.

Let $\{U_\lambda\}_{\lambda \in \Lambda}$ be an indexed family of open sets. Then note

X \setminus \bigcup_{\lambda \in \Lambda} U_\lambda = \bigcap_{\lambda \in \Lambda} ~ (X \setminus U_\lambda).

Since each $U_\lambda$ is open, it follows that the left-hand side is an arbitrary intersection of finite sets, which is finite. Thus the arbitrary union of open sets is open.

Let $U_1, \dots, U_n$ be finitely many open sets. Then note

X \setminus \bigcap_{i = 1}^n U_i = \bigcup_{i = 1}^n ~ (X \setminus U_i).

Since each $U_i$ is open, it follows that the left-hand side is a finite union of finite sets, which is finite. Thus the finite intersection of open sets is open. Thus $\Tau_f$ is a well-defined topology.

[2.1.6]Definition(Finer, Coarser, and Comparable Topologies)#

Suppose $X$ is a set with two topologies, $\Tau$ and $\Tau'$ . If $\Tau \subseteq T'$ , then $\Tau$ is coarser than $\Tau'$ . If $\Tau \ne \Tau'$ , then $\Tau$ is strictly coarser than $\Tau'$ . Conversely, $\Tau'$ is finer or strictly finer than $\Tau$ . If $\Tau \subseteq \Tau'$ or $\Tau' \subseteq T$ , then we say the two topologies are comparable.