THE COMPACTNESS OPERATOR IN GENERAL TOPOLOGY

This chapter discusses the compactness operator in general topology. The role of (bi)compactness has increased tremendously during the past half century. The chapter focuses on the strengthening of this notion at the expense of the Hausdorff property. It discusses a few special cases of importance such as: (1) ϱ = ɛ holds exactly for those topological spaces in which the compact sets coincide with the closed sets. (2) ϱ 2 = ɛ. In this case ϱ and ɛ form a group of order 2 with ɛ as the identity. Spaces supplied with such a minus topology are called antispaces. These are exactly those spaces in which the square-compact subsets coincide with the closed subsets. The locally compact Hausdorif spaces and the metrizable spaces are for example antispaces.