Characterizations of certain topological structures by means of well-ordered open coverings
Sconyers, Woodlea Bernard
Sconyers, Woodlea Bernard
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1968
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Abstract
In 1967 J. Mack initiated a study of paracompact spaces as related to well-ordered open coverings. Mack characterized paracompact spaces by the property that every well-ordered open covering has a locally finite open refinement. It may be natural to determine if the condition of local finiteness in the above description can be replaced by such weaker conditions as the closure preserving property or the cushioned property in the same way as in the usual characterizations of paracompact spaces due to Michael. However, it is unknown if the closure preserving union of compact spaces is paracompact. In this case every well-ordered open covering has a closure preserving closed refinement. Therefore, it would be worthwhile to attempt to obtain the weakest possible refinement property which would characterize paracompact spaces in terms of well-ordered open coverings. The main theorem is concerned with a new type of refinement property, called linearly hereditarily closure preserving families. It is shown that a normal space X is paracompact if and only if for every well-ordered open covering of X, there is a linearly hereditarily closure preserving open covering whose closure forms a refinement. Since a hereditarily closure preserving closed covering of a space X determines the weak topology for X in the sense of Morita, the notion of linearly hereditarily closure preserving families may be useful in obtaining a generalization of a well-known theorem originally concerned with the paracompactness of CW-complexes. From this idea, the notion of the pseudo-weak topology is introduced and the paracompactness of a space having the pseudo-weak topology with respect to a closed covering of paracompact subspaces is established. Various other topological properties are characterized by means of well-ordered open coverings. The main problem concerning the Lindelof spaces is the equivalence of linearly Lindelof spaces and Lindelof spaces. Some partial results on this equivalence are given. It may be concluded that the notion of well-ordered open coverings may be of value in placing new light on the descriptions of topological properties. It should be noted that many of the proofs of those characterizations given in this paper depend heavily on the technique of induction on cardinality. This fact may be suggestive of the peculiar feature of this type of characterization of topological properties.
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Topology
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Dissertation
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iii, 54 leaves, bound
Department
Mathematics