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What does vanishing at infinity mean for a topological space?

  1. Oct 3, 2013 #1
    If X is a locally compact Haussorff space, then the set of continuous functions of compact support form a normal vector space C_c(X) with the supremum norm, and the completion of this space is the space C_0(X) of functions vanishing at infinity, i.e. the space of functions f such that f can be contintuous extended to a function g on the one point compactification of X and g(∞) = 0.

    However, this definition is rather abstract, and requires investigating continuous extensions on the one-point compactification, something that you wouldn't expect for a notion as simple as a limit at infinity. So is there a more practical characterization of vanishing at infinity, something that is directly in terms of X?

    Any help would be greatly appreciated.

    Thank You in Advance.

    EDIT: I found exactly the characterization I was looking for in "en.wikipedia.org/wiki/Vanish_at_infinity" [Broken]. So now the question becomes, how would you prove that the definition given in the Wikipedia article is equivalent to the one I gave above?
     
    Last edited by a moderator: May 6, 2017
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  3. Oct 3, 2013 #2

    Erland

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    This follows easily if we consider the topology for the one point compactification. What does a neigbourhood of the point at infinity look like?
     
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