Uniform continuity and normality of metric spaces in ZF

• Autorzy: Keremedis K.

Identyfikatory

DOI

10.4064/ba8122-10-2017

• Języki publikacji: EN

Abstrakty

EN

Let X = (X, d) and Y = (Y, ρ) be two metric spaces. (a) We show in ZF that: (i) If X is separable and f: X → Y is a continuous function then f is uniformly continuous iff for any A, B ⊆ X with d(A, B) = 0, ρ (f(A), f(B)) = 0. But it is relatively consistent with ZF that there exist metric spaces X, Y and a continuous, nonuniformly continuous function f : X → Y such that for any A, B ⊆ X with d(A, B) = 0, ρ (f(A), f(B)) = 0. (ii) If S is a dense subset of X, Y is Cantor complete and f : S → Y a uniformly continuous function, then there is a unique uniformly continuous function F : X → Y extending f. But it is relatively consistent with ZF that there exist a metric space X, a complete metric space Y, a dense subset S of X and a uniformly continuous function f : S → Y that does not extend to a uniformly continuous function on X. (iii) X is complete iff for any Cauchy sequences (x_n)_n∈N and (y_n)_n∈N in X, if [wzór] then d({x_n : n ∈ N},{y_n : n ∈ N}) > 0. (b) We show in ZF+CAC that if f : X → Y is a continuous function, then f is uniformly continuous iff for any A, B ⊆ X with d(A, B) = 0, ρ (f(A), f(B)) = 0.

Słowa kluczowe

EN

axiom of choice; continuous function; uniformly continuous function; complete; Cantor complete; separable metric spaces

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2017
• Tom: Vol. 65, no. 2
• Strony: 113--124
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Keremedis K.

• Department of Mathematics, University of the Aegean, Karlovassi, Samos 83200, Greece

Bibliografia

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• [7] K. Keremedis and E. Tachtsis, Products of some special compact spaces and restricted forms of AC, J. Symbolic Logic 75 (2010), 996-1006.
• [8] S. Kundu and T. Jain, Atsuji spaces: equivalent conditions, Topology Proc. 30 (2006), 301-325.
• [9] S. Mrówka, On normal metrics, Amer. Math. Monthly 72 (1965), 998-1001.
• [10] J. R. Munkres, Topology, Prentice-Hall, Englewood Cliffs, NJ, 1975.
• [11] M. E. Rudin, A new proof that metric spaces are paracompact, Proc. Amer. Math. Soc. 20 (1969), 603.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).