Imbedding of a compact Hausdorff space

[radou]

Homework Statement

This is a short question, just to check.

Let X be a compact Hausdorff space, and suppose that for each x in X, there exists a neighborhood U of x and a positive integer k such that U can be imbedded in R^k. One needs to show that there exists a positive integer N such that X can be imbedded in R^N.

The Attempt at a Solution

Now, basically I don't see the difference between the proof of this fact and that of Munkres' Theorem 36.2., Section 36, except (of course) that the integer N will be different. But the same proof should work, unless I'm mistaken. Am I right on this one?

 

[micromass]

It seems that you are correct. The same proof applies!

 

[radou]

It seems that you are correct. The same proof applies!

OK, thanks!