# Reverse mathematics and the strong Tietze extension theorem

#### Paul Shafer, University of Leeds. Part of the proofs, constructions and computations seminar series.

In second-order arithmetic, the Tietze extension theorem can be phrased by asserting that if X is a complete separable metric space, C is a closed subset of X, and f is a continuous and bounded function from C to the reals, then there is a continuous and bounded extension F of f to all of X. This version of the Tietze extension theorem is known to be provable in RCA_0. Giusto and Simpson introduced what they called the strong Tietze extension theorem, in which X is required to be compact and f and F are required to be uniformly continuous (in the sense of having moduli of uniform continuity). Giusto and Simpson showed that WKL_0 suffices to prove the strong Tietze extension theorem but that RCA_0 does not, which lead them to conjecture that the strong Tietze extension theorem is equivalent to WKL_0 over RCA_0. We confirm this conjecture.

Paul Shafer, University of Leeds

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