Finite-to-one restrictions of continuous functions
H. Toruńczyk (1985)
Fundamenta Mathematicae
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H. Toruńczyk (1985)
Fundamenta Mathematicae
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Roman Sikorski (1951)
Fundamenta Mathematicae
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M. Charalambous (1976)
Fundamenta Mathematicae
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Alois Švec (1968)
Czechoslovak Mathematical Journal
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Andrzej Komisarski (2006)
Fundamenta Mathematicae
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Let X be a compact Hausdorff topological space. We show that multiplication in the algebra C(X) is open iff dim X < 1. On the other hand, the existence of non-empty open sets U,V ⊂ C(X) satisfying Int(U· V) = ∅ is equivalent to dim X > 1. The preimage of every set of the first category in C(X) under the multiplication map is of the first category in C(X) × C(X) iff dim X ≤ 1.
Keiô Nagami (1981)
Fundamenta Mathematicae
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Charles Herladns (1984)
Fundamenta Mathematicae
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Miroslav Katětov (1995)
Commentationes Mathematicae Universitatis Carolinae
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Using certain ideas connected with the entropy theory, several kinds of dimensions are introduced for arbitrary topological spaces. Their properties are examined, in particular, for normal spaces and quasi-discrete ones. One of the considered dimensions coincides, on these spaces, with the Čech-Lebesgue dimension and the height dimension of posets, respectively.