felix hausdorff

Commentary on [H 1933a]
V. Kanovei; P. Koepke
This note was written after the memoir [KL 1932], which dealt with different
aspects of 6s-operations. In particular KANTOROVITCH and LiVENSON consider
a modification of the 6s-operation on planar sets that applies differently to
different cross-sections of planar sets.
More precisely, consider subsets of a product space of the form Z = X x Y.
Suppose that (y? is a map that associates a 6s-operation ^y over the index
set INI with any y ^ Y. Let Xn C X for any n. Define Ocp({Xn}nGN) to
be the set C C X x Y such that, for any y E Y, the cross-section Cy —
{x : (x,2/) G C} coincides with $^({Xn}nGN)- This can be called: a crosssection-wise
6s-operation.
Suppose further that 21 is a system of subsets X, while C^ is a system
of subsets oi Z = X xY. KANTOROVITCH and LIVENSON define ([KL 1932],
p. 250) € to be projective rel. 21 if there is an assignment if as above such that
C is equal to the collection (1(^(21) of all sets of the form C = ^^p{{Xn}ne^)',
where {X^lnGN is an arbitrary sequence of sets in 21.
HAUSDORFF proposes the following modification of this concept, perhaps
distilled from the "Fundamental Theorem On Projections" in [KL 1932], p. 264.
Say that a system € as above is 8s-projective rel. 21 if and only if for any 6soperation
^ there exists a 6s-operation ^ such that the class ^21 of all subsets
of X obtained by the action of $ on sets in 21 is equal to the class TT'^C of
all X-projections of those subsets of X x Y obtained by the action of ^ on
sets in €.
HAUSDORFF'S theorems I, H and ni are present in [KL 1932]. In particular,
theorem H which states that if € is projective rel. 2t then (t is 6s-projective rel.
21, is the content of the "Fundamental Theorem" on p. 264. However HAUS­
DORFF'S exposition is simpler and more focused, and he also removes certain
restrictions in [KL 1932]. For instance, the condition that y is a compact space,
as in the cited "Fundamental Theorem", is removed. HAUSDORFF'S demonstration
does not use any topological properties of Y at all.
The final Theorem IV in HAUSDORFF'S note asserts that if X is arbitrary
and Y separable (more exactly, 2nd countable) then the system of all open
subsets of X X y is projective rel. the collection of all open subsets of X.
(By Theorem III, we can replace both occurrences of "open" by "closed".) The
proof is remarkably simple: if {l^^jneN is a base for the topology of y, then
let Py = {n:yeVn} and $^(^1,^2,.-.) =[JnePy^ri-
References
[KL 1932] KANTOROVITCH, L.; LIVENSON, E.: Memoir on the analytical
operations and projective sets, I. Fundamenta Math. 18 (1932), 214-279.
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