nLab Taimanov theorem


Let XX be a T 1T_1 space, i:DXi\colon D \hookrightarrow X a dense subspace, YY a compact Hausdorff space and f:DYf\colon D \to Y be any continuous map. If for all disjoint closed A,BYA, B \subseteq Y we have i(f 1(A))¯i(f 1(B))¯=\overline{i(f^{-1}(A))} \cap \overline{i(f^{-1}(B))} = \emptyset, then there is a continuous extension of ff to XX.

There is a variant where XX is arbitrary and YY is T 3T_3.


  • A.D. Taĭmanov, О распространении непрерывных отображений топологических пространств (On extension of continuous mappings of topological spaces.) Mat. Sbornik N.S. 31(73), (1952). 459–463. page

The result is stated, for instance, as Theorem 3.2.1 in

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