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

Sierpinski space is <</A>53#>$T_0$ http://planetmath.org/encyclopedia/T0.html but not <</A>54#>$T_1$ http://planetmath.org/encyclopedia/T1.html. It is $T_0$ because $\lbrace x\rbrace$ 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 $\mathcal{T}=P(U)\cup \lbrace X\rbrace$