Tietze Extension Theorem

**k-misako** · August 26th 2011, 01:59 AM

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!

MK

**FernandoRevilla** · August 26th 2011, 07:20 AM

Originally Posted by k-misako

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.

Simply think that is good to know that we can extend (continuously) a continuous map. For example, $(\mathbb{R},T_u)$ is a normal space and for $A=[a,b]$ closed set and $f:A\to \mathbb{R}$ continuous, it is very intuitive to see that it is possible to extend continuously $f$ to $\mathbb{R}$ . However, consider $B=(0,1]$ (not closed) and the continuous map $f:B\to \mathbb{R}$ , $g(x)=1/x$ . Also, it is intuitive to see that we can't extend continuously $g$ to $\mathbb{R}$ .

**Drexel28** · August 26th 2011, 09:38 AM

Originally Posted by k-misako

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!

MK

To add to what Dr. Revilla said, the ostensible reason for this theorem is that it proves a 'comfort theorem' for Tychnoff spaces. A 'comfort theorem' is often an embedding theorem which says that something super abstract is really not so abstract after all. For example, in algebra there are simple examples of comfort theorems: every group can be embedded into a symmetric group (Cayley's theorem), the fact that every unital ring can be embedded into the endomorphism ring of an abelian group, the fact that every integral domain can be embedded into a field, etc. Thus, all three of these abstract structures (groups, unital rings, integral domains) are made less abstract since, at least intuitively, they are subgroups of $S_n$ , subrings of $\text{End}(A)$ , or subrings of $k$ (for a field $k$ ) respectively. There are also examples of these 'comfort theorems' which are also useful for proving theorems that, without this 'extra comfort', would be hard to prove. While the third example I mentioned is an example of this (i.e. the fact that $R\hookrightarrow \text{Frac}(R)$ can be used to prove that if $R$ is an infinite integral domain then the canonical map $R[x]\hookrightarrow R^R$ is an embedding) probably the best example of a 'functional comfort theorem' can be found in topology. Namely, the fact that every locally compact Hausdorff space embeds into a compact Hausdorff space (this is the Alexandroff compactification) enables one to prove, for example, that every locally compact Hausdorff space is Tychnoff (something not easy to do otherwise, or at least not as easy as when you know the Alexandroff compactification exists: if $X$ is locally compact Hausdorff and $X_\infty$ denotes it's Alexandroff compactification then we know that $e:X\xrightarrow{\approx}e(X)\subseteq X_\infty$ , but $e(X)$ is Tychonoff since $X_\infty$ is Tychnoff (it's normal even) and being Tychonoff passes to subspaces).

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 $X$ is Tychonoff then $X$ embeds into $[0,1]^J$ for some indexing set $J$ . Thus, every Tychonoff space (a really quite abstract object) is secretly just a subspace of a cube!

I hope that helps.

**Drexel28** · August 26th 2011, 09:46 AM

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 $X$ is a locally compact Hausdorff space then any positive linear functional $\varphi$ on $C_c(X)$ are of the form $\displaystyle \varphi(f)=\int_X f(x)d\mu(x)$ for a unique Borel measure $\mu$ .

**FernandoRevilla** · August 26th 2011, 10:06 AM

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.

**k-misako** · August 28th 2011, 04:49 AM

Thank you very much, both of you!
These answers really help.
Especially, I love a mathematician's (or pre-mathematician's?) informal statements such as #3, although I cannot understand them at all NOW.
Thanks a lot!

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