Characterization of the function classes C(Y)|X

Á Császár, E. Makai

Research output: Contribution to journalArticle

Abstract

Let X be a topological space and let Φ ⊂ C(X). Then there exists a topological space Y containing X as a subspace and such that Φ = C(Y)|X, if and only if Φ is weakly composition closed, i.e., for any index set I, any fi ∈ Φ (i ∈ I) and any continuous map k:RI → R we have k ○ 〈fi〉 ∈ Φ, where 〈fi〉:X → RI is the map with i-th coordinate fi. The analogous statement is valid for functions to any T1 space, rather than to R, and even we can consider functions to any set of T1 spaces, and then a generalization of the above statement is valid, with a suitably defined weak composition closedness property. We also show that some earlier results on characterization of function classes Φ ⊂ C(X) of the form C(Y)|X, with Y some extension of a given topological space X, and on the characterization of function classes C(〈X, T〉), with T some topology on a given set X, respectively, can be generalized in an analogous way as above, by means of composition properties analogous to the above one or by filter closedness (for functions to any set of T3 spaces, or to any set of topological spaces, respectively).

Original languageEnglish
Pages (from-to)95-107
Number of pages13
JournalActa Mathematica Hungarica
Volume81
Issue number1-2
DOIs
Publication statusPublished - Oct 1998

ASJC Scopus subject areas

  • Mathematics(all)

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