Spaces determined by countably many locally compact subspaces

Abstract

The theory of compactly generated spaces, alternatively k-spaces, plays an important role in general and algebraic topology. In this paper, we develop a theory of compact generation for non-Hausdorff spaces using locally compact spaces. We are particularly interested in the case that countably many locally compact spaces suffice and restrict our attention to that case. The notion of ℓcω-spaces (which can be considered as a type of kω-spaces) is introduced, which are determined by countably many locally compact spaces. It is proved that the product space of two ℓcω-spaces is still an ℓcω-space. We slightly modify the notion of an ℓcω-space to apply it to posets equipped with the Scott topology. The notions of ℓcω-posets, ℓfω-posets and c-posets are introduced. We develop a corresponding theory for these three kinds of posets and investigate the conditions under which the Scott topology on the product of two posets is equal to the product of the individual Scott topologies and under which the Scott topology on a dcpo is sober. Several such conditions are given.