Locally compact spaces
Category: Science > Math
Asked by: moosebeard-ga
List Price: $5.00
09 May 2004 16:18 PDT
Expires: 08 Jun 2004 16:18 PDT
Question ID: 343706
Let X be any locally compact space and let Y be any space. Can you find a continuous function, f, taking X into Y such that the image f(X) is not locally compact? I just want an example, for some spaces X and Y and some function f. Please keep in mind that f is not an open map, otherwise the image is locally compact by a trivial argument.
Re: Locally compact spaces
Answered By: mathtalk-ga on 09 May 2004 16:48 PDT
Hi, moosebeard-ga: Here's a Web page that gives examples of both locally compact and not locally compact topologies: [PlanetMath - examples of locally compact and not locally compact spaces] http://planetmath.org/encyclopedia/ExamplesOfLocallyCompactAndNotLocallyCompactSpaces.html Let Y be your favorite topological space which is not locally compact, for example an infinite-dimensional normed vector space. Let X be the same point set as Y but with the discrete topology, so that trivially X is locally compact. Let f:X -> Y be the mapping induced by the underlying identity of point sets. Since the inverse image under f of any subset of Y is open in X because of the discrete topology there, f is trivially continuous. By the choice of space Y, the image f(X) = Y is not locally compact. regards, mathtalk-ga
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