The Tietze extension theorem is a fundamental result in topology, and it has several equivalent statements that provide different perspectives on the theorem. Here are some equivalent statements to Tietze's extension theorem:
Extension from Closed Subsets: Any real-valued, continuous function defined on a closed subset of a normal topological space can be extended to a continuous function on the whole space. This is the core statement of the theorem and is described on Wikipedia.
Target Space Replacement: The target space [0,1] in the original statement can be replaced with any closed interval [a, b]. This is mentioned in a lecture note by Zuoqin Wang available here.
Extension with Specific Properties: For every continuous function f defined on closed subsets F1 and F2 of a normal space X, there exists a continuous function f*:X→[a,b] such that f* restricted to F1 is equal to f restricted to F1 and f* restricted to F2 is equal to f. This is discussed on Math Stack Exchange.
Generalization to Polish Spaces: If X is a Polish space (even a normal space) and Y=R^n, then a continuous function f:C→Y on a closed set C can be extended to a continuous function on the whole space. This is a topic explored on Math Overflow.
Reverse Mathematics Context: The Tietze extension theorem can also be viewed in the context of reverse mathematics, where it is shown that certain versions of the theorem are equivalent over RCA0. This is detailed in a paper available here.
These equivalent statements provide a deeper understanding of the theorem's implications and its applications in various areas of mathematics.