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# Sierpinski space

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

Sierpinski space is $T_{0}$ but not $T_{1}$. 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 $\mathcal{T}=P(U)\cup\{X\}$.

Keywords:

topology

Related:

T1Space, T2Space, SeparationAxioms

Synonym:

Sierpi\'nski space

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

54G20*no label found*

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