What's so cool about local compactness?

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?

• AFAIK, local compactness + Hausdorff allows you to develop/extend methods of calculus, using Haar measures. Commented Nov 26, 2013 at 5:49
• Locally compact groups are the objects on which we can do analysis Commented Nov 26, 2013 at 5:49
• Specifically, Harmonic Analysis. Commented Nov 26, 2013 at 6:18
• It's like compactness, but just locally. And compactness is the beeswax of topology! Commented Nov 26, 2013 at 15:03
• 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. Commented Nov 26, 2013 at 15:09

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).