Glasnik Matematicki, Vol. 51, No. 2 (2016), 447-452.

CERTAIN WEAKLY GENERATED NONCOMPACT, PSEUDO-COMPACT TOPOLOGIES ON TYCHONOFF CUBES

Leonard R. Rubin

Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, USA
e-mail: lrubin@ou.edu

Abstract.   Given an uncountable cardinal ℵ, the product space Iℵ, I=[0,1], is called a Tychonoff cube. A collection of closed subsets of a subspace Y of a Tychonoff cube Iℵ that covers Y determines a weak topology for Y. The collection of compact subsets of Iℵ that have a countable dense subset covers Iℵ. It is known from work of the author and I. Ivanšić that the weak topology generated by this collection is pseudo-compact. We are going to show that it is not compact. The author and I. Ivanšić have also considered weak topologies on some other ``naturally occurring'' subspaces of such Iℵ. The new information herein along with the previous examples will lead to the existence of vast naturally occurring classes of pseudo-compacta any set of which has a pseudo-compact product. Some of the classes consist of Tychonoff spaces, so the product spaces from subsets of these are also Tychonoff spaces.

2010 Mathematics Subject Classification.   54A10, 54B10, 54D30.

Key words and phrases.   First uncountable ordinal space, products, pseudo-compact, Tychonoff cube, weak topology.

Full text (PDF) (free access)

DOI: 10.3336/gm.51.2.11

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