Topological properties related to the upper bound topology

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Linda Gentry Rapp (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site:
Hughes Hoyle

Abstract: If X is a set with topologies S and T, the upper bound topology for X is the set T[S,T] defined as follows: a set U c X is in T[S,T] provided if x € U there is an s € S and a t € T such that x € s n t c U. In this paper we examine the upper bound topology and its relation to the topologies S and T. In Chapter I we look at the separation axioms. We find that if (X,S) is T0, T1, or Hausdorff, then (X,T[S,T]) has the same property. It is also observed that the implication is not reversible. Further, it is shown that if (X,T[S,T]) is either regular, normal, Urysohn, or metric then no conclusion can be drawn about (X,S). In Chapter II we look at countability and find that if (X,S) and (X,T) are both first countable or second countable then (X,T[S,T]) has the same property. It is also seen that if (X,T[S,T]) is separable then (X,S) is separable. Examples are given to show that the implications are not reversible. In Chapter III it is observed that if (X,T[S,T]) is either compact, countably compact, Lindelof, or connected then so is (X,S). Examples are given to show that no conclusion can be drawn about locally connected. The reader is expected to have a working knowledge of point-set topology and is referred to [l], [2], and [3] for definitions and theorems not covered in this paper.

Additional Information

Language: English
Date: 1972
Topological spaces

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