TIETZE Extension Theorem Metric
 

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Re: TIETZE Extension Theorem Metric
 

Here I am giving you the TIETZE Extension Theorem Metric in detail

(Tietze extension theorem). A space X is T4 if and only if
for every closed subset A and every continuous function f : A → R, there is a function g : X → R extending f, meaning g|A = f.

Proof sketch. A rigorous proof of this theorem requires quoting some basic results from analysis, so we give a sketch instead. The “if” direction is easy and is omitted. For the “only if” direction, we compose f with a
homeomorphism R → (−1, 1) to get a bounded function. Assuming f is bounded, we approximate f by a function f1 defined on X, using the lemma.
The difference is presumably nonzero, so we approximate the difference by another function f2 and then f1 + f2 is a function on X that restricts to a better approximation of f. Repeating this process infinitely many times,
the error of the approximations goes to zero exponentially fast, and the approximations converge to the desired function g (this uses the fact from analysis that uniform limits of continuous functions are continuous).
More on metric spacesSome last comments about separation and countability properties as they
pertain to metric spaces. As we mentioned before, metric spaces are sepa-rable if and only if they are second countable.