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I would like to consult with anyone who is reading this post on how do you explain the distinction between compact spaces and locally compact spaces to students who had just completed topology course and is now in functional analysis class. In topology, the syllabus cover only Compact spaces and in Functional analysis class, the students were being approach to locally compact spaces.

I can't explain very well and I just say that for locally compact spaces, it is like a subset of a set where the set is a compact spaces. In other words it is a more specific as compared to compact spaces and I don't feel my response is quite clear. Hope to seek some help on the explanation. Thank You.

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In a locally compact space, each point has a neighborhood with compact closure. The space is compact if and only if we can choose it independently of the point we are considering (the real line with the usual topology is a good way to see this). In functional analysis, I think you have seen the theorem which states that a normed space over $\Bbb R$ or $\Bbb C$ is locally compact if and only if it's finite dimensional (which is equivalent to compactness of the unit ball). –  Davide Giraudo Aug 3 '12 at 10:36
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Compact spaces are small in a whole, locally compact in the neighborhood of each point. Locally compact spaces can be "big" in a whole, but their local properties are good enough. –  Norbert Aug 3 '12 at 11:01
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Note, also, that a locally compact Hausdorff space is very nearly compact, in the sense that it is possible to add a single point to the space and make this larger space compact. –  Arthur Fischer Aug 3 '12 at 11:32
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@Samdra Here's a good example to consider: $\Bbb{R}$ with the Euclidean topology is not compact but locally compact. On the other hand a space like $[0,1]$ that is compact is also locally compact. While $\Bbb{Q}$ with the euclidean topology cannot be locally compact because it is not a Baire Space. –  user38268 Aug 3 '12 at 13:09
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up vote 2 down vote accepted

You also need to be careful, as there are two possible and used definitions of locally compact: (1) every point has a compact neighbourhood; (2) every point has a base of compact neighbourhoods. The advantages of the second definition are: (i) it is in the spirit of definitions of local properties, and (ii) it is very often this property which is used in practice. The two definitions agree for Hausdorff spaces.

Of course the previous comments are on the ball.

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