Properties of cocontinuous functions and cocompact spaces

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.