# A question about open subsets of Hilbert space whose complements are compact sets

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.

• 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. – 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 $$xzy$$.
• 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). – Pietro Majer Feb 26 '15 at 23:05