Give an example of a $\sigma$-compact complete metric space which is not locally compact. A space $X$ is said to be locally compact if for each $x\in X$, there exist an open set $U$ and a compact subset $K$ of $X$ such that $x\in U\subseteq K$.


1 Answer 1


Let $\mathcal H$ be a separable Hilbert space with basis $v_1,v_2,\ldots$ and let $X$ be the closed subspace of all vectors that can be written as $tv_n$ with $t\in[0,1]$ and $n\in\mathbb N$. Then $X$ is $\sigma$-compact, since it is the union of the compact sets $K_n = \{tv_n \mid t\in [0,1]\}$. But $X$ is not locally compact, because there is no compact neighbourhood of 0.

  • 5
    $\begingroup$ I believe your space with an hedgehog space of spininess $\aleph_0$. $\endgroup$
    – Wojowu
    Dec 24, 2021 at 12:23
  • 2
    $\begingroup$ Yes it is indeed! =) $\endgroup$
    – Squala
    Dec 24, 2021 at 12:25

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