felix hausdorff

Commentary on [H 1934]
H. Herrlich, M. Husek, G. Preuss
The paper consists of three parts having a common theme, namely the relationship
between open mappings and properties related to completeness.
The first part concerns preserving completeness by open mappings. As is
mentioned in the paper, SlERPINSKi proved a special case of Theorem I in [Sie
1930]. The precise statement of SIERPINSKI'S result is as follows:
Si E est un ensemble Gs (d'un espace a m dimensions) et si f{x) est une fonction
definie et continue sur E qui transforme tout ensemble ouvert relativement
a E en un ensemble ouvert relativement a f{E), f{E) est aussi un ensemble
Gs. ([Sie 1930], p. 173).
That is why HAUSDORFF speaks about SlERPlNSKl's result for subsets of
Euclidean spaces (ENGELKING in his book [Eng 1989] says that SIERPINSKI
proved his result for separable metric spaces).
HAUSDORFF proved the result for general metric spaces in his notes from
29.1.1931 (Kapsel 35, Fasz.407). Why the result was published only in 1934
is explained in the second paragraph of [H 1934]. These notes also contain an
example (not included in the final publication) which explains that the result is
not valid for multi-valued mappings: X is the set of reals, Y the set of rationals
and / assigns to a: in X all the i/'s in Y such that |a: — i/| < 1; then f{G) is
open in Y for every open G in X, and f~^{H) is open in X for every open H
inF.
HAUSDORFF's procedure is based on a characterization of topological completeness
by means of the so-called "closed basis". That characterization was
proved independently by VEDENISOV and HAUSDORFF (see Commentary to [H
1924]).
HAUSDORFF's proof is elegant and elementary and is used even today, if one
cannot go outside metric spaces. Otherwise, procedures used by MICHAEL for
paracompact spaces are used. MICHAEL generalized Theorem I in [Mic 1959]
as follows:
A paracompact continuous and open image of a completely metrizable space is
completely metrizable. In fact, instead of completeness of the domain one can
assume completeness of fibers f~^{x).
The second part concerns the following result of MAZURKIEWICZ on extensions
of open mappings (a solution of ARONSZAJN'S problem from [Aro 1931])
published in [Maz 1932]:
Soient R, T deux espaces complets, separables, A d R, f une transformation
interieure de A, f{A) — B