TITLE: $\sigma$ - compact and locally compact metric space
QUESTION [8 upvotes]: 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

REPLY [7 votes]: 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 $i$.