nLab Taimanov theorem

Statement

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

There is a variant where $X$ is arbitrary and $Y$ is $T_3$.

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

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