nLab
Taimanov theorem

Statement

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.

References

  • 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. mathnet.ru page

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

Last revised on September 25, 2019 at 07:53:15. See the history of this page for a list of all contributions to it.