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Q: Locally compact spaces ( Answered 5 out of 5 stars,   0 Comments )
Subject: Locally compact spaces
Category: Science > Math
Asked by: moosebeard-ga
List Price: $5.00
Posted: 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.
Subject: Re: Locally compact spaces
Answered By: mathtalk-ga on 09 May 2004 16:48 PDT
Rated:5 out of 5 stars
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]

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
moosebeard-ga rated this answer:5 out of 5 stars

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