# Tikhonov space

A topological space in which every finite set is closed and such that for every closed set $P$ and any point $x$ not in $P$ there exists a continuous real-valued function $f$ on the whole space taking the value 0 at $x$ and the value 1 at every point of $P$. The class of Tikhonov spaces coincides with the class of completely-regular $T_1$-spaces (cf. Completely-regular space). In a Tikhonov space any two distinct points can be separated by disjoint neighbourhoods (in other words, the Hausdorff separation axiom is satisfied), but not every Tikhonov space is normal (cf. Normal space). A.N. Tikhonov (1929) characterized Tikhonov spaces as subspaces of compact Hausdorff spaces.