### 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 f_{i} ∈ Φ (i ∈ I) and any continuous map k:R^{I} → R we have k ○ 〈f_{i}〉 ∈ Φ, where 〈f_{i}〉:X → R^{I} is the map with i-th coordinate f_{i}. The analogous statement is valid for functions to any T_{1} space, rather than to R, and even we can consider functions to any set of T_{1} 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 T_{3} spaces, or to any set of topological spaces, respectively).

Original language | English |
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Pages (from-to) | 95-107 |

Number of pages | 13 |

Journal | Acta Mathematica Hungarica |

Volume | 81 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Oct 1998 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Acta Mathematica Hungarica*,

*81*(1-2), 95-107. https://doi.org/10.1023/a:1006523112596