[Math] $\sigma$ – compact and locally compact metric space

Question

Is the following sentence is true?

Each complete, separable and $\sigma$ – compact metric space is locally compact.

I suppose (but I'm not sure) it is a truth, becouse it was evidently used in the paper of Łukasz Stettner "Remarks on Ergodic Conditions of Markov Processes on Polish Spaces"(108 p.) which I am studyng now.

full text of this work – http://www-bcf.usc.edu/~lototsky/InfDimErg/Stettner-InfDimMarkProc.pdf

Answer 1

For a counterexample let $e_i, i=1, 2, \ldots$ be the standard unit vectors in $\ell^2$, and $X$ the union of the line segments $L_i$ joining 0 to $e_i$ for all $i$.