Sierpinski space

Sierpinski space is the topological space $X=\{x,y\}$ with the topology given by $\{X,\{x\},\emptyset\}$ .

Sierpinski space is $T_{0}$ (http://planetmath.org/T0) but not $T_{1}$ (http://planetmath.org/T1). It is $T_{0}$ because $\{x\}$ is the open set containing $x$ but not $y$ . It is not $T_{1}$ because every open set $U$ containing $y$ (namely $X$ ) contains $x$ (in other words, there is no open set containing $y$ but not containing $x$ ).

Remark. From the Sierpinski space, one can construct many non- $T_{1}$ $T_{0}$ spaces, simply by taking any set $X$ with at least two elements, and take any non-empty proper subset $U\subset X$ , and set the topology $\mathcal{T}$ on $X$ by $\mathcal{T}=P(U)\cup\{X\}$ .

Title	Sierpinski space
Canonical name	SierpinskiSpace
Date of creation	2013-03-22 12:06:26
Last modified on	2013-03-22 12:06:26
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	9
Author	CWoo (3771)
Entry type	Definition
Classification	msc 54G20
Synonym	Sierpiński space
Related topic	T1Space
Related topic	T2Space
Related topic	SeparationAxioms