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As I study more algebraic number theory, I hear more and more often about local compactness: locally compact fields, locally compact topological groups, Stone-Čech compactification of locally compact spaces, etc. etc.

In one paper, I even read that proving what the characters of $\Bbb {Q_p, R}$ are is trivial using properties of locally compact spaces.

So I am wondering, what are all those nice properties, and what kind of (counter-)examples should I keep in mind when thinking about local compactness?

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    $\begingroup$ AFAIK, local compactness + Hausdorff allows you to develop/extend methods of calculus, using Haar measures. $\endgroup$ Commented Nov 26, 2013 at 5:49
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    $\begingroup$ Locally compact groups are the objects on which we can do analysis $\endgroup$ Commented Nov 26, 2013 at 5:49
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    $\begingroup$ Specifically, Harmonic Analysis. $\endgroup$ Commented Nov 26, 2013 at 6:18
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    $\begingroup$ It's like compactness, but just locally. And compactness is the beeswax of topology! $\endgroup$
    – Asaf Karagila
    Commented Nov 26, 2013 at 15:03
  • $\begingroup$ In some of your examples, compactness is too strong a requirement, and local compactness is the next best thing you can hope for: no infinite topological field is compact. $\endgroup$ Commented Nov 26, 2013 at 15:09

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Every locally compact Hausdorff space is Tychonoff, hence it has "enough" continuous functions. This is used extensively. A prominent example is the proof of Gelfand duality: If we associate to a locally compact Hausdorff space $X$ the $C^*$-algebra $C(X)$ of complex continuous functions, then we obtain an anti-equivalence of categories between locally compact Hausdorff spaces and commutative $C^*$-algebras (with suitably defined morphisms).

Technical but important: If $Y$ is locally compact, then for all spaces $X,Z$ the map $$C(X,C(Y,Z)) \to C(X \times Y,Z)$$ is well-defined and bijective (exponential law). The special case $Y=[0,1]$ shows that a homotopy between continuous maps $X \to Z$ is really just a map from $X$ to the path space $C(I,Z)$.

Quite related to the exponential law: If $Y$ is locally compact, then $Y \times -$ preserves quotient maps (this doesn't hold for arbitrary $Y$, although many topologists use this, perhaps having in mind a convenient category of spaces).

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