topological space

A topological space is a set $X$ together with a set $\mathcal{T}$ whose elements are subsets of $X$ , such that

• $\emptyset \in \mathcal{T}$
• $X \in \mathcal{T}$
• If $U_j \in \mathcal{T}$ for all $j \in J$ , then $\bigcup_{j \in J} U_j \in \mathcal{T}$
• If $U \in \mathcal{T}$ and $V \in \mathcal{T}$ , then $U \cap V \in \mathcal{T}$

Elements of $\mathcal{T}$ are called open sets of $X$ . The set $\mathcal{T}$ is called a topology on $X$ . A subset $C \subset X$ is called a closed set if the complement $X \setminus C$ is an open set.

A topology $\mathcal{T}'$ is said to be finer (respectively, coarser) than $\mathcal{T}$ if $\mathcal{T}' \supset \mathcal{T}$ (respectively, $\mathcal{T}' \subset \mathcal{T}$ ).

Examples

• The discrete topology is the topology $\mathcal{T} = \mathcal{P}(X)$ on $X$ , where $\mathcal{P}(X)$ denotes the power set of $X$ . This is the largest, or finest, possible topology on $X$ .
• The indiscrete topology is the topology $\mathcal{T} = \{\emptyset, X\}$ . It is the smallest or coarsest possible topology on $X$ .
• Subspace topology
• Product topology
• Metric topology

Bibliography

1

J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.

2

J. Munkres, Topology (2nd edition), Prentice Hall, 1999.