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$\mathbb{R}^n\not\cong\mathbb{R}^n\setminus\{0\}$ are not homeomorphic is a triviality from Algebraic Topology. On the other hand, if $X$ is an infinite dimensional Banach space, then $X \cong X\setminus\{0\}$.

Question. In the $\infty$-dimensional spaces, which sets are "removable" like $\{0\}$ in $X$? Can you give an explicit homeomorphic map in such a case?

T. Amdeberhan
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    Wow, $X \simeq X \setminus {0}$, really? Do you have a reference for that? Or is getting the reference part of the question? – Vanessa Feb 03 '17 at 21:26
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    V. Klee, Two topological properties of topological linear spaces, Israel J. Math. 4 (1964), 211-220. – T. Amdeberhan Feb 03 '17 at 21:28
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    I suggest that the title should be changed to better reflect the question – Yemon Choi Feb 03 '17 at 22:05
  • I changed the title. Feel free to edit further if you don't like it. I thought $\setminus$ would be clearer since $-$ is ambiguous in a vector space. – Nate Eldredge Feb 03 '17 at 22:45
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    There is a huge literature on this. Start with MR0250342
    Henderson, David W. Infinite-dimensional manifolds are open subsets of Hilbert space. Topology 9 1970 25–33.     – Bill Johnson Feb 03 '17 at 22:56
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    See also http://mathoverflow.net/questions/254521/producing-non-trivial-homotopy-classes-in-infty-dimensional-vector-spaces-by/254550?noredirect=1#comment626776_254550 and references therein – Thomas Rot Feb 04 '17 at 01:50
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    Many know but let me still write that there is entire well-developed branch devoted to this kind of questions. This branch is the infinite dimensional topology. There were pioneering Scottish Book questions by Karol Borsuk. You may call it pre-history. Then there were early pioneers: Anderson, Bessaga, Henderson, Klee, Szankowski, ... Then there were the very advanced, outstanding results by Chapman, Toruńczyk, West, followed by R. D. Edwards, etc. – Włodzimierz Holsztyński Feb 05 '17 at 05:34
  • Van Mill 's book "Prerequisites and an introduction" (see https://www.amazon.com/Infinite-Dimensional-Prerequisites-Introduction-North-Holland-Mathematical/dp/0444871330) shows that for separable Banach spaces $X$ ,$X$ is homeomorphic to $X \setminus A$, where $A$ is a Z-set (defined in the book), Z-sets include all $\sigma$-compact subsets of $X$. I was quite suprised when I first learnt this.. There won't be easy explicit homeomorphisms though. For ${0}$ there is, see Bessaga and Pelczynski's fine book. – Henno Brandsma Feb 09 '17 at 02:31

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