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

•
$\mathrm{\varnothing}\in \mathcal{T}$

•
$X\in \mathcal{T}$

•
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 ${\mathcal{T}}^{\prime}$ is said to be finer (respectively, coarser^{}) than $\mathcal{T}$ if ${\mathcal{T}}^{\prime}\supset \mathcal{T}$ (respectively, ${\mathcal{T}}^{\prime}\subset \mathcal{T}$).
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 $\mathcal{T}=\{\mathrm{\varnothing},X\}$. It is the smallest or coarsest possible topology on $X$.
 •
 •
 •
References
 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
 2 J. Munkres, Topology (2nd edition), Prentice Hall, 1999.
Title  topological space 
Canonical name  TopologicalSpace 
Date of creation  20130322 11:49:52 
Last modified on  20130322 11:49:52 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  12 
Author  djao (24) 
Entry type  Definition 
Classification  msc 2200 
Classification  msc 5500 
Classification  msc 5400 
Synonym  topology 
Related topic  Neighborhood^{} 
Related topic  MetricSpace 
Related topic  ExamplesOfCompactSpaces 
Related topic  ExamplesOfLocallyCompactAndNotLocallyCompactSpaces 
Related topic  Site 
Defines  open 
Defines  closed 