Ditopological Compactness
by
Maroof Gohar
Hacettepe University, Ankara
Coauthors: Lawrence M. Brown (Hacettepe University)

By a texturing of a set S we mean a subset S of the powerset P(S) of S which is a point separating complete, completely distributive lattice with respect to inclusion, which contains S and \emptyset and for which meets coincide with intersections and finite joins coincide with unions. The pair (S, S) is then called a texture space or simply texture. The simplest example of a texture in (S, P(S)). Another important example is (L, L), where L=[0, 1] and L={[0, r), [0, r] | r in [0, 1]}.

Since a texturing of S need not be closed under set complementation we replace the notion of topology by that of ditopology. Here (\tau, \kappa) is a ditopology if \tau subset or equal S contains S and \emptyset, and is closed under arbitrary joins and finite meets, while dually \kappa subset or equal S contains S and \emptyset, and is closed under arbitrary meets and finite joins. We assume no relation between the family \tau of open sets and the family \kappa of closed sets in general. A bitopology on S gives rise to a ditopology on (S, P(S)) by taking \tau to be one of the topologies and \kappa the closed sets for the second topology. A natural ditopology on (L, L) above is \tau = {[0, r) | r in [0, 1]} \cup {L}, \kappa = {[0, r] | r in [0, 1]} \cup {\emptyset}.

The set A in S is compact if whenever A subset or equal \/ {G_i : i in I}, G_i in \tau, we have J subset or equal I finite with A subset or equal \cup {G_j : j in J}, and cocompactness of A is defined dually. (S, S, \tau, \kappa) is called compact if S is compact and stable if all A in S\{S} are compact. Cocompactness and costability are defined dually. A ditopological texture space may be called dicompact if it is compact, cocompact, stable and costable. For ditopologies on (S, P(S)), dicompactness corresponds to bitopological joint (join) compactness, the above definition reflecting R. D. Kopperman's subdivision of this concept into compactness and stability properties [2]. It is trivial that the ditopological texture space (L, L, \tau, \kappa) above is dicompact.

Some characterizations of dicompactness may be found in [1]. In this paper we present further properties of dicompactness. In particular we generalize the characterization of compactness given in [3] to the case of ditopological compactness and cocompactness, and hence to dicompactness.

References

1. L. M. Brown and M. Diker, Ditopological texture spaces and intuitionistic sets, Fuzzy Sets and Systems 98, 1998, 217-224.
2. R. D. Kopperman, Asymmetry and duality in topology, Topology and its Applications 66, 1995, 1-39.
3. S. Mrówka, Compactness and product spaces, Coll. Math. 7, 1959, 19-22.

Date received: August 3, 2001