Closed-constructible functions are piecewise closed

Abstract

Dedicated to the memory of topologist and friend D. Ranchin MSC: primary 54C10, 28A05, 03E15, 54E40, 26A21A subset of a topological space X is constructible if it belongs to the smallest algebra of subsets that contains all open subsets of X. We prove that if a continuous function f : X → Y between separable metrizable spaces maps closed subsets of X onto constructible sets of Y , then X admits a countable closed cover C such that for each C ∈ C the restriction f |C is closed.The present paper continues the series of publications about decomposibility of Borel functions 1 f between separable metrizable spaces into a finite or countable number of open, closed and continuous functions.In the first publication [7] we have raised the question about preservation of complete metrizability by some simplest Borel functions. The author has limited his first work to only one question about the preservation of completeness, since it was not clear whether such a decomposition was possible. It is known, however, that if a decomposition is possible, it leads to preservation of even other Borel classes [9]. Recently, this question was fully and affirmatively resolved by Gao and Kleinfield [1], Holický and Pol [3].In the following publications [8,10] we have obtained such a decomposition into open, closed and continuous functions in the case of simplest Borel functions.In the present paper we give an affirmative solution for the first case when images of closed sets 2 are unrestricted combinations of closed and open sets.

Open-constructible functions

“…The present paper continues the series of publications about decomposibility of Borel functions [6], [8] -see also [2], [3] where functions of such type are the main subject.…”

“…Analogously, using H-sets we can easy extend the proof for closed-constructible functions in [6] to the case of closed-resolvable.…”