Abstract
Generally, the term uc-ness means some continuity is uniform. A metric space X is uc when any continuous function fromX to [0, 1] is uniformly continuous and a metrizable space X is a Nagata space when it can be equipped with a uc metric. We consider natural forms of uc-ness for the \({\omega_\mu}\)-metric spaces, which fill a very large and interesting class of uniform spaces containing the usual metric ones, and extend to them various different formulations of the metric uc-ness, by additionaly proving their equivalence. Furthermore, since any \({\omega_\mu}\)-compact space is uc and any uc \({\omega_\mu}\)-metric space is complete, in the line of constructing dense extensions which preserve some structure, such as uniform completions, we focus on the existence for an \({\omega_\mu}\)-metrizable space of dense topological extensions carrying a uc \({\omega_\mu}\)-metric. In this paper we show that an \({\omega_\mu}\)-metrizable space X is uc-extendable if and only if there exists a compatible \({\omega_\mu}\)-metric d on X such that the set X′ of all accumulation points in X is crowded, i.e., any \({\omega_\mu}\)-sequence in X′ has a d-Cauchy \({\omega_\mu}\)-subsequence in X′.
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References
E. M. Alfsen and O. Njastad, Totality of uniform structures with linearly ordered base, Fund. Math., 52 (1963), 253–256.
G. Artico, U. Marconi and J. Pelant, On supercomplete \(\omega _\mu \)-metric spaces, Bull. Pol. Acad. Sci., 44 (1996), 299–310.
G. Artico and R. Moresco, \(\omega _\mu \)-additive topological spaces, Rend. Sem. Mat. Univ. Padova, 67 (1982), 131–141.
M. Atsuji, Uniform continuity of continuous functions in metric spaces, Pacific J. Math., 8 (1958), 11–16.
M. Atsuji, Uniform continuity of continuous functions on uniform spaces, Canadian J. Math., 13 (1961), 657–663.
G. Beer, Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proceedings of the Amer. Math. Soc., 95 (1985), 653–658.
G. Beer, More about metric spaces on which continuous functions are uniformly continuous, Bull. Austral. Math. Soc., 33 (1986), 397–406.
G. Beer, Topologies on closed and closed convex sets, Kluwer (Dordrecht, 1993).
G. Beer, UC spaces revisited, American Math. Monthly, 95 (1988), 737–739.
L. W. Cohen and C. Goffman, The topology of ordered abelian groups, Trans. Amer. Math. Soc., 67 (1949), 310–319.
A. Di Concilio, Proximity: a powerful tool in extension theory, function spaces, hyperspaces, boolean algebra and point-set topology, in: Beyond Topology, F. Mynard and E. Pearl (eds.), Contemporary Mathematics Amer. Math. Soc., 486 (2009), 89–114.
A. Di Concilio and S. A. Naimpally, Atsuji spaces, continuity versus uniform continuity, in: Proc. VI Brazilian Topology Meeting, Unicamp Brasil (1988), pp. 30–40.
R. Engelking, General Topology, Heldermann (1989).
M. Forti and F. Honsell, Models of self-descriptive set theories, in: Partial Differential Equations and the Calculus of Variations, Volume I, Essays in Honor of Ennio De Giorgi, Birkhauser (Boston Inc., 1989).
J. R. Isbell, Uniform Spaces, Mathematical Surveys, Amer. Math. Soc., 12 (1964).
T. Jain and S. Kundu, Atsuji completions vis-á-vis hyperspaces, Math. Slovaca, 58 (2008), 497–508.
T. Jain and S. Kundu, Atsuji spaces: equivalent conditions, Topology Proc., 30 (2006), 301–325.
R. J. Malitz, Set theory in which the axiom of foundation fails, Ph.D. Thesis, UCLA (Los Angeles, 1976) (unpublished).
U. Marconi, Some conditions under which a uniform space is fine, Comment. Math. Univ. Carolin., 34, (1993), 543-547.
A. A. Monteiro and M. M. Peixoto, Le nombre de Lebesgue et la continuité uniforme, Portugaliae Mathematica, 10 (1951).
S. G. Mrowka, On normal metrics, Amer. Math. Monthly, 72 (1965), 998–1001.
J. Nagata, On the uniform topology of bicompactifications, J. Inst. Polytech. Univ., 1 (1950), 28–38.
S. A. Naimpally and B. D. Warrack, Proximity Spaces, Cambridge Tract No. 59 (1970).
S. A. Naimpally, Proximity Approach to Problems in Topology and Analysis, Oldenburg Verlag (München, 2009).
P. Nyikos and H. C. Reichel, On uniform spaces with totally ordered bases II, Fund. Math., 93 (1976), 1–10.
J. Rainwater, Spaces whose finest uniformity is metric, Pacific J. Math., 9 (1959), 567–570.
H. C. Reichel, Some results on uniform spaces with linearly ordered bases, Fund. Math., 98 (1978), 25–39.
Sierpiński W., Sur une propriété des ensembles ordonnés, Fund. Math., 36 (1949), 56–57.
R. Sikorski, Remarks on some topological spaces of high power, Fund. Math., 37 (1950), 125–136.
F. W. Stevenson and W. J. Thron, Results on \(\omega _\mu \)-metric spaces, Fund. Math., 65 (1969), 317–324.
A. H. Stone, Metrizability of decomposition spaces, Proc. Amer. Math. Soc., 7 (1956), 690–700.
G. Toader, On a problem of Nagata, Mathematica (Cluj), 20 (1978), 78–79.
W. Waterhouse, On uc spaces, Amer. Math. Monthly, 72 (1965), 634–635.
S. Willard, General Topology, Addison-Wesley (1970).
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Di Concilio, A., Guadagni, C. Uniform continuity in \({\omega_\mu}\)-metric spaces and uc \({\omega_\mu}\)-metric extendability. Acta Math. Hungar. 150, 153–166 (2016). https://doi.org/10.1007/s10474-016-0643-9
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DOI: https://doi.org/10.1007/s10474-016-0643-9
Key words and phrases
- Nagata extension
- uniform space
- uniform space with a totally ordered base
- \({\omega_\mu}\)-metric space
- uc-ness in \({\omega_\mu}\)-metric space
- \({\omega_\mu}\)-additive topological space
- Hausdorff–Bourbaki uniformity
- Hausdorff convergence
- Vietoris topology
- pseudo-Cauchy net