topological space

A 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 on $X$. A subset $C\subset X$ is called a if the complement $X\setminus C$ is an open set.

A topology $\mathcal{T}^{\prime}$ is said to be finer (respectively, ) 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 $\mathcal{T}=\{\emptyset,X\}$. 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 2013-03-22 11:49:52 Last modified on 2013-03-22 11:49:52 Owner djao (24) Last modified by djao (24) Numerical id 12 Author djao (24) Entry type Definition Classification msc 22-00 Classification msc 55-00 Classification msc 54-00 Synonym topology Related topic Neighborhood Related topic MetricSpace Related topic ExamplesOfCompactSpaces Related topic ExamplesOfLocallyCompactAndNotLocallyCompactSpaces Related topic Site Defines open Defines closed