What are some uses of Stone–Čech compactification? What is the motivation for introducing this notion?
Most textbooks on topology construct Stone–Čech compactification and prove that this construction satisfies a particular universal property (which essentially sais that it is the left adjoint to the inclusion functor from the category of compact Hausdorff to the category of topological spaces).
But this seems to be quite isolated from the rest of (algebraic) topology. For instance, Munkres' book on topology has a short section on the Stone–Čech compactification, but after that section he doesn't use the Stone–Čech compactification anywhere else in the book, as far as I can see.
Wikipedia gives one application of the Stone–Čech compactification: it can be used to construct the dual space of the bounded sequences of reals.
I wonder: That's all??! Probably not! What's the real motivation and use of Stone–Čech compactification?
Best Answer
I'd say the original motivation was to show that every Tychonoff space $X$ had a compactication (a compact Hausdorff space in which $X$ is densely embedded) and the natural construction that people came up with (via Tychonoff cubes $[0,1]^I$) turned out to have very nice special properties: this compactification is the maximal one for $X$ (all other conceivable compactifications of $X$ are quotients of it), has nice function extension properties and so had (as we'd now say) functoriality.
Of course for discrete spaces it occurs as a Boolean space for power set algebras (another natural road) and so is part of nice theory and duality via an algebraic road.
It also occurred naturally when people started studying rings of continuous functions $C(X)$ for Tychonoff spaces, its points correspond to the maximal ideals of such rings.
There are also applications: van der Waerden's theorem can be proved by considering the compact semigroup $\beta \Bbb N$. And as $\ell^\infty \simeq C(\beta \Bbb N)$ it allows for a slightly more concrete representation of the dual of $\ell^\infty$ as a space of measures on a compact space $\beta \Bbb N$, which is an interesting space in its own right (see the Handbook of General Topology which has a whole chapter on this one space).