Splittability, T_\delta, T_\zeta and T_A

by

Alan Hanna and Brian McMaster

Proceedings of the Third Galway Topology Colloquium (1999)

The power and usefulness of cleavability (also known as splittability) have been well established within the framework of topology by A. V. Arhangel'skii and his associates. A property P is preserved by splitting if, whenever a space X is splittable over a collection of P spaces, then X is a P space. Not all separation axioms are preserved by splitting but, for those which are not, a positive result can usually be obtained by demanding further properties of the maps that do the splitting (e.g. that they should be closed as well as continuous). A few axioms, notably T_A and T_\delta, resist this approach. We illustrate Alan Hanna's recent method for dealing with such recalcitrant axioms, which presupposes a lower degree of separation to hold in X. In addition, we make some fresh observations on T_\zeta and its splittability behaviour.

Keywords: splittable, partially ordered set, low separation.
MSC: 06A06, 54C99.

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Date received: November 7, 1999.