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In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem or
Urysohn-Brouwer lemma) states that any continuous, real-valued function on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.
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The usual statement of the theorem assumes a bounded codomain, and yields an extension with the same codomain. The (possibly) unbounded version is also correct, but the bounded formulation makes explicit something useful that isn't stated in the unbounded version, namely that the extended function
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that invoked this theorem as part of the proof. I don't know if invoking it was really needed or not. I don't know if that counts as some "other branch of mathematics".
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continuous function on a closed subset can be extended, regardless of it's co-domain. However, the formal statement clearly states the the function's co-domain must be
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can be chosen to have the same bounds. This page would be more useful if this fact were added. (I will not do so, however, at least anytime soon.)
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How useful is this theorem to other branches of mathematics, or within functional analysis. This can be theoretical or practical. Cheers.
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made the required change 8 minutes after you posted the above note.
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The opening line of the article seems to imply that
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