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 $x\in U\subseteq K$.
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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 $\aleph_0$. $\endgroup$– WojowuDec 24, 2021 at 12:23

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