Universal elements for some classes of spaces

D. N. Georgiou, Stavros Iliadis, A. C. Megaritis

Abstract

In the paper two dimensions, denoted by dm and Dm, are defined in the class of all Hausdorff spaces. The dimension Dm does not have the universality property in the class of separable metrizable spaces because the family of all such spaces X with Dm(X) < 0 coincides with the family of all totally disconnected spaces in which there are no universal elements. In we gave the dimension-like functions dmK,B E and DmK,B E, where K is a class of subsets, E a class of spaces and B a class of bases and we proved that in the families P(dm K, B E < K)and P(Dm K, B E < K) of all spaces X for which dm K,B E (X) < K and Dm K, B E (X) < K, respectively there exist universal elements. In this paper, we give some new dimension-like functions and define using these definitions classes of spaces in which there are universal elements.

Keywords

Dimension-like function; Saturated class; Universal space

Subject classification

54B99; 54C25

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References

R. Engelking, Theory of dimensions, finite and infinite, Sigma Series in Pure Mathematics, 10. Heldermann Verlag, Lemgo, 1995. viii+401 pp.

S. D. Iliadis, Universal spaces and mappings, North-Holland Mathematics Studies, 198. Elsevier Science B.V., Amsterdam, 2005. xvi+559 pp.

D. N. Georgiou, S. D. Iliadis and A.C. Megaritis, Dimension-like functions and universality, Topology Appl. 155 (2008), 2196-2201. https://doi.org/10.1016/j.topol.2007.05.024

A. K. O'Connor, A new approach to dimension, Acta Math. Hung. 55, no. 1-2 (1990), 83-95. https://doi.org/10.1007/BF01951390

R. Pol, There is no universal totally disconnected space, Fund. Math. 79 (1973), 265-267. https://doi.org/10.4064/fm-79-3-265-267

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