Topological space

Abstractly, a topological space $(𝑋, T)$ consists of a set $𝑋$ and a collection of subsets $T \subseteq P (𝑋)$ such that¹ #m/def/topology

$T$ contains at least $\emptyset$ and $𝑋$ .
Any finite or infinite union of subsets in $T$ is also in $T$ .
Any finite intersection of subsets in $T$ is also in $T$ .

where $T$ is called a topology on $𝑋$ , and is said to contain open subsets of $𝑋$ . A subset of $𝑋$ is called closed iff its compliment is open. Thus, in any topological space $(𝑋, T)$ the subsets $𝑋$ and $\emptyset$ are clopen².

On any set $𝑋$ we can easily form the Discrete topology $P (𝑋)$ (every set is clopen) and the Trivial topology ${\emptyset, 𝑋}$ .
Two topologies on the same space $𝑋$ can be compared in terms of Coarseness and fineness of topologies.
A topology can be generated by a Topological basis.

Properties

#state/tidy | #SemBr | #lang/en

2020, Topology: A categorical approach, §0.1, p. 1 ↩
Simultaneously open and closed. ↩

Topological space

Properties

Footnotes