Let \(X\) be a topological space. Two subsets \(S, T\subseteq X\) are separated by neighborhoods if there exist disjoint open sets \(U\supseteq S\) and \(V\supseteq T\).
\(X\) is \(T_0\) if for all \(x\neq y\) in \(X\) there exists an open set containing exactly one of \(x\) and \(y\).
\(X\) is \(T_1\) if for all \(x\neq y\) in \(X\) there exists an open set containing \(x\) and not \(y\).
\(X\) is \(T_2\), or Hausdorff, if for all \(x\neq y\) in \(X\) the sets \(\ts{x}\) and \(\ts{y}\) are separated by neighborhoods.
\(X\) is \(T_{2\frac12}\) if for all \(x\neq y\) in \(X\) there exist open sets \(U\ni x\) and \(V\ni y\) with \(\cl_X(U)\intersect\cl_X(V) = \emptyset\).
\(X\) is \(T_3\) if it is \(T_0\) and regular: for every \(x\in X\) and every closed \(F\subseteq X\) with \(x\notin F\), the sets \(\ts{x}\) and \(F\) are separated by neighborhoods.
\(X\) is \(T_{3\frac12}\) if it is \(T_0\) and completely regular: for every \(x\in X\) and every closed \(F\subseteq X\) with \(x\notin F\) there exists a continuous \(f\colon X\to[0,1]\) with \(f(x) = 0\) and \(f(F)\subseteq\ts{1}\).
\(X\) is \(T_4\) if it is \(T_1\) and normal: any two disjoint closed subsets of \(X\) are separated by neighborhoods.
A space is \(T_1\) if and only if every singleton is closed.
[@Mun00].
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D-OM7TDSeparation axioms $T_0$ through $T_4$variant-of