On the space of quotient objects of compact Hausdorff spaces
by
Kateryna Koporkh
Pre-Carpathian University, Ivano-Frankivsk, Ukraine

Let X be a compact Hausdorff space. Following E.Shchepin [1] we say that two continuous onto maps f_i: X→ Y_i, i=1, 2, are equivalent (written f₁ ~ f₂) provided that there exists a homeomorphism h: Y₁→ Y₂ such that f₂=hf₁ (here Y₁, Y₂ are also compact Hausdorff spaces).

The set F(X)={[f] | f is an onto map of compact Hausdorff spaces } of equivalence classes can be topologized in different ways. This can be done by identifying every [f] ∈ F(X), for f: X→ Y, with the set of f_*(C(Y)) ∈ CL(C(X)), where CL(C(X)) stands for the set of nonempty closed subsets of the Banach space C(X) of the continuous functions on X, and to topologize F(X) with one of different topologies on the space CL(C(X)). In the sequel, we consider the Wijsman topology on the set CL(C(X)) (see [2]).

It turns out, however, that this construction is not functorial on the category Comp of compact Hausdorff spaces.

However, if i: X→ Y is an a embedding of X as an open subset of Y, then the induced map F(X)→ F(Y) is continuous. The proof of this is based on the fact that the natural map m: F(X₁)×F(X₂)→ F(X₁∪X₂) is continuous.

By F₀(X) we denote the set of all equivalence classes of open subsets of X. Every element [f] of the set F₀(X) is identified with the family {f^-1(f(x)) | x ∈ X} ∈ exp²(X) and is topologized by the Vietoris topology on exp²(X).

We prove that this construction is functorial on the category Comp₀ of compact Hausdorff spaces and open surjective maps.

[1] E.V. Shchepin, Functors and uncountable powers of compacta, Uspekhi mat. nauk, 36(1983), no.3, 3-62.

[2] G.Beer, Topologies on Closed and Closed Convex Sets, Kluwer, Dordrecht, 1993. p. 340.

Date received: May 9, 2008