In the book "Real and Complex Analysis" by Rudin, he often uses the condition that a space is locally compact Hausdorff in order to present results in a general manner. The thing is, I'm not very used to such condition. Most books of analysis/measure-theory that I've read present results in terms of metric spaces/separable/complete.
Thus, I was wondering if there is a precise relation between such notions. For example, does locally compact Hausdorff implies completeness or separability? Is the opposite implication true?
Take for example the following theorem by Rudin:
If $X$ is locally compact Hausdorff, and $\mu$ a measure on the borelians of $X$. Then for $1\leq p < \infty$, $C_c(X)$ is dense in $L^p(\mu)$.
Now, I was wondering if this theorem could be somehow stated, but something like, if $X$ is Polish, then this is true. Hence, what I'm really interested in is to know if there is a way to somehow relate this type of spaces. If the implications are not true, is there an extra condition that tie them together?
Best Answer
Both implications fail.
Product space $[0,1]^A$ with $A$ uncountable is compact Hausdorff but not separable and not metrizable.
Hilbert space $l_2$ is complete separable metric, but not locally compact.
Of course, many common spaces have both properties. Indeed, an open subset of $\mathbb R^n$ is completely metrizable separable locally compact Hausdorff.