Spaces determined by countably many locally compact subspaces

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 $\ell c_{\omega }$-spaces (which can be considered as a type of $k_{\omega }$-spaces) is introduced, which are determined by countably many locally compact spaces. It is proved that the product space of two $\ell c_{\omega }$-spaces is still an $\ell c_{\omega }$-space. We slightly modify the notion of an $\ell c_{\omega }$-space to apply it to posets equipped with the Scott topology. The notions of $\ell c_{\omega }$-posets, $\ell f_{\omega }$-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.