Would you please give me some good examples which enable me to appreciate the power (or wonder) of Tietze Extension Theorem? I prefer easier examples which require no advanced knowledges if possible.
Thank you for reading this question, and I am looking forward to your replies!
So, how does Tietze's extension theorem imply a 'comfort theorem'? Well, while the above may be a little advanced for your tastes, the following should be no problem for you. Namely, Tietze's theorem implies that if is Tychonoff then embeds into for some indexing set . Thus, every Tychonoff space (a really quite abstract object) is secretly just a subspace of a cube!
I hope that helps.
Ah! Perhaps 'requiring too much advanced knowledge' I forgot to mention that Tietze's extension theorem is a huge part in proving arguably the most famous and important (at least in analysis) 'comfort theorem'--the general form of the Riesz representation theorem: if is a locally compact Hausdorff space then any positive linear functional on are of the form for a unique Borel measure .
Well Drexel28, you have provided a thesis . Only I'd like to comment that although the Tietze Extension Theorem is "almost" a direct consequence of the Urysohn lemma, we only need the latter for proving the Riesz Representation Theorem.