Tikhonov theorem

on the compactness of products

The topological product of an arbitrary set of compact spaces (cf. Compact space) is compact. This is one of the fundamental theorems in general topology. It was established by A.N. Tikhonov in 1929. It plays a very essential role, and often a key one, in constructions in all branches of general topology and in many of its applications. In particular, it is of fundamental importance for the construction of compactifications of completely-regular $T_1$-spaces (that is, Tikhonov spaces). Using it one can construct the Stone–Čech compactification of an arbitrary Tikhonov space. Tikhonov's theorem allows one to exhibit standard compact spaces — the generalized Cantor discontinuum $D^\tau$, which is the product of discrete two-point sets indexed by $\tau$, and the Tikhonov cube $I^\tau$, which is the product of $\tau$ copies of the unit interval on the real line. Here any cardinal number can be taken for $\tau$. The importance of the generalized Cantor set $D^\tau$ and the Tikhonov cube $I^\tau$ are related, moreover, to the fact that these are universal objects: Every zero-dimensional $T_2$-compactum is homeomorphic to a closed subspace of some $D^\tau$, and every $T_2$-compactum is homeomorphic to a closed subspace of some $I^\tau$.

Tikhonov's theorem is applied in the proof of the non-emptiness of an inverse limit of compact spaces, in constructing the theory of absolutes, and in the theory of compact groups. If one takes into account its indirect applications, then almost all of general topology lies within the sphere of influence of this theorem. It is equally difficult to list its direct and indirect applications in other branches of mathematics. They arise practically everywhere where the notion of compactness plays an important role — in particular, in functional analysis (Banach spaces with the weak topology, measures on topological spaces), in the general theory of optimal control, etc.

References

[1]	J.L. Kelley, "General topology" , Springer (1975)
[2]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)

References

[a1]	R. Engelking, "General topology" , Heldermann (1989)
[a2]	A.N. Tikhonov, "Über die topologische Erweiterung von Räumen" Math. Ann. , 102 (1930) pp. 544–561