Let be compact. Prove or disprove that is separable in general.Problem:

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- Mar 27th 2010, 07:23 PMDrexel28Topology
Let be compact. Prove or disprove that is separable in general.__Problem:__ - Mar 28th 2010, 03:25 AMtonio
- Mar 28th 2010, 08:25 AMDrexel28
- Mar 28th 2010, 09:10 AMDrexel28
__Proof:__

I didn't use the whole Stone-Weierstrass theorem, only the Weierstrass.

**Lamma:**Let be the set of all polynomials in with rational coefficients. Then, is dense in

**Proof:**Let be arbitrary. Choose such that .

Now, the mapping given by is continuous and so it assumes a maximum .

Choose such that .

Then, . It follows that .

Thus,

.

It follows that is dense in .

It follows that since given by is a surjection that is countable.

Now, for every there exists, by Tietze's extension theorem, an extension such that .

Also, for every there exists some such that . So, let and . Clearly and thus countable. Clearly then, is also countable.

Also, let and be arbitrary. Letting and the subsequent be as before. Then, and .

It follows that is a countable dense subset of . The conclusion follows.