Sierpinski space

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

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 $𝒯$ on $X$ by $𝒯=P(U)\cup \{X\}$.