# 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 SierpinskiSpace 2013-03-22 12:06:26 2013-03-22 12:06:26 CWoo (3771) CWoo (3771) 9 CWoo (3771) Definition msc 54G20 Sierpiński space T1Space T2Space SeparationAxioms