[Munkres Topology] 12. Topological Spaces

 This article is one of the posts in the Textbook Commentary Project.

Definition. A topology on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ having the following properties:

(1) $\varnothing\in \mathcal{T}$, $X\in \mathcal{T}$.

(2) $\forall \{U_\alpha\}_\alpha$ s.t. $U_\alpha \in \mathcal{T}$, then $\bigcup_\alpha U_\alpha \in \mathcal{T}$.

(3) $\forall \{U_i\}_{i=1}^n$ s.t. $U_i \in \mathcal{T}$, then $\bigcap_{i=1}^n U_i \in \mathcal{T}$.

A set $X$ for which a topology $\mathcal{T}$ has been specified is called a topological space.

If $X$ is a topolgocial space with topology $\mathcal{T}$, we say that a subset $U$ of $X$ is an open set of $X$ is $U$ belongs to the collection $\mathcal{T}$.

Definition. Suppose that $\mathcal{T}$ and $\mathcal{T}'$ are two topologies on a given set $X$. If $\mathcal{T}'\supset \mathcal{T}$, we say that $\mathcal{T}'$ is finer than $\mathcal{T}$; if $\mathcal{T}'$ properly contains $\mathcal{T}$, we say that $\mathcal{T}'$ is strictly finer than $\mathcal{T}$. We also say that $\mathcal{T}$ is coarser than $\mathcal{T}'$ or strictly coarser, in these two respective situations. We say $\mathcal{T}$ is comparable with $\mathcal{T}'$ is either $\mathcal{T}'\supset \mathcal{T}$ or $\mathcal{T}\supset \mathcal{T}'$.