I was curious about the relations between second-countable, separable, Lindelöf and $\sigma$-compact topologies in the context of metric spaces.
I am aware of the following implications in general topological spaces:
- second-countable $\Rightarrow$ separable $\not \Rightarrow$ Lindelöf, $\;$ [thanks bof]
- $\sigma$-compact $\Rightarrow$ Lindelöf
- second-countable + locally compact $\Rightarrow$ $\sigma$-compact
as well as the reversed implications in the case of metric spaces:
- Lindelöf $\Leftrightarrow$ separable $\Leftrightarrow$ second-countable
Since all the proofs I've seen so far require the LC condition I assume it is not true in general that second-countable topological spaces are $\sigma$-compact (although seeing an actual counterexample would be nice).
So what about metrizable topological spaces?
Ideas so far:
If we can proof that every subset of a $\sigma$-compact space is again $\sigma$-compact, then this would follow from the fact, that every separable metric space is homeomorphic to a subset of the Hilbert cube (which is compact). $\;$[debunked by bof]