Topological homogeneity. Topological groups and their continuous images. (English. Russian original) Zbl 0642.54017

Arkhangel’skij, A. V.

Russ. Math. Surv. 42, No. 2, 83-131 (1987); translation from Usp. Mat. Nauk 42, No. 2(254), 69-105 (1987).

A space X is (topologically) homogeneous if for all x,y\(\in X\) there exists a homeomorphism of X onto itself that takes x to y. Examples of homogeneous spaces are topological groups, the Cantor set, the Hilbert cube, the pseudo-arc, etc. Homogeneity plays a central role in the solution of several outstanding topological problems such as the problem of the topological characterization of the Hilbert cube, etc. So far, no systematic study of homogeneity has been made. This valuable survey is as far as I know the first attempt to study homogeneity in its own right. The author discusses two closely connected themes: topologically homogeneous spaces and topological invariants of topological groups. In addition, he states numerous open problems which will undoubtedly create much activity among topologists.

Reviewer: J.van Mill

MSC:

54C99	Maps and general types of topological spaces defined by maps
54-02	Research exposition (monographs, survey articles) pertaining to general topology

Keywords:

cardinal invariants; topological homogeneity; Vietoris topology; hyperspaces; products; retracts; predensity; existence of \(G_{\delta }\)-points; precharacters; continuous images of topological groups