Let $H$ be an infinite-dimensional separable Hilbert space. Let $C$ be the intersection of a denumerably infinite sequence of sets, each of which is the complement of a compact subset of $H$. In other words (equivalently), let $\ C\ $ be a complement in $\ H\ $ of a $\sigma$-compact set (where a $\sigma$-compact set is simply a countable union of compacta in $\ H$).

QUESTION: Is $C$ necessarily connected and locally connected?

The answer is clearly "NO", if $H$ is a finite dimensional Euclidean space.

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    $\begingroup$ In each infinite dimensional separable Frechet space $X$ the complement of any $\sigma$-compact subset is homeomorphic to $X$, and in particular, it is connected and locally connected. This is a standard fact of infinite dimensional topology proved in the 60s by (I think) R.D.Anderson. A reference is e.g. Theorem V.6.4 in the book "Selected topics in infinite dimensional topology" by Bessaga and Pelczynski. $\endgroup$ – Igor Belegradek Mar 1 '15 at 13:11

General fact: if we remove a countable collection $K_1,K_2,\ldots$ of compact sets from an infinite-dimensional Banach space $X$, the remaining set $V$ is locally and globally path-connected (actually any open ball is path-connected).

Proof. If $K$ is a compact subset of $X$, and $x\in X$ is a point, then the set union $\cup_{z\in K} xz$ of segments $xz$ is a compact (it is the image of the compact set $[0,1]\times K\subset \mathbb{R}\times X$ under continuous map $(t,z)\to x+t(z-x)$.) Let $x$, $y$ be two points in $V$. The set $\Omega$ of points $z$ for which the segment $zx$ or $zy$ contains a point from some $K_i$ is a countable union of compact sets, thus has Baire first category. Therefore there exists a point $z\notin \Omega$ and $x$, $y$ may be joined by a path $xzy$.

  • $\begingroup$ Nice! We may also say that the radial projection from $x$ onto the hyperplane $H$ (which is a continuous map $X\setminus\{x\}\to H$) takes the countable union of compact subsets $X\setminus V$ to a countable union of compact subsets therefore of first category in $H$. (Also, actually you have proven that $V$ is locally path connected and connected). $\endgroup$ – Pietro Majer Feb 26 '15 at 23:05
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    $\begingroup$ Sorry, I always say 'Banach space' instead of 'infinite-dimensional Banach space'. Fixed. $\endgroup$ – Fedor Petrov Feb 26 '15 at 23:55
  • $\begingroup$ Many thanks for the proof which I was not quite able to find for myself $\endgroup$ – Garabed Gulbenkian Feb 27 '15 at 20:48
  • $\begingroup$ V is a G-delta set (an intersection of a countable sequence of open sets). Since you have shown V to be both connected and locally connected, it follows that V is also arc-wise connected. $\endgroup$ – Garabed Gulbenkian Feb 28 '15 at 21:28
  • $\begingroup$ In my case it'd cause a math overflow. :) $\endgroup$ – Włodzimierz Holsztyński Feb 28 '15 at 22:19

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