In this paper we study the concept of cotopology in the areas of cocon·tinuous functions and cocompact spaces. Initially we investigate and provide needed results concerning closed bases for a topological space. We then study cocontinuous functions by relating them to various other weaker forms of continuous functions, namely c-continuous, almost continuous and weakly continuous. We show that if (Y,U) is locally compact T₂, then f:(X,T)-->(Y,U) is cocontinuous if and only if f⁻¹(0) ε T for every 0 ε U such that (Y - 0) is compact. We note that every almost continuous function is cocontinuous, and we provide conditions under which a weakly continuous function is cocontinuous. We also show that a cocontinuous function from a saturated space to a regular space is continuous.
In the area of cocompact spaces we first provide a partial answer to a question of J. M. Aarts as to when the union of cocompact subsets of a space is cocompact. We show that the union of a closed cocompact subset and a closed compact subset is cocompact. We then introduce the properties, locally cocompact and somewhere cocompact, and relate them to property L which was introduced by R. McCoy. We show that every somewhere cocompact regular space has property L, and that every locally cocompact regular space has property L locally. We provide examples to show that neither cocompact nor locally cocompact is equivalent to property L. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/74672 |
| Date | January 1973 |
| Creators | Francis, Gerald L. |
| Contributors | Mathematics |
| Publisher | Virginia Polytechnic Institute and State University |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | iii, 78 leaves., application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 22409151 |
Page generated in 0.0018 seconds