Characterization of T0 spaces for liminf convergence being topological

Abstract

The concepts of S-convergence and liminf convergence are extended from dcpos to T0 topological spaces. We introduce WC-spaces, a weakening of C-spaces, which were introduced as the generalization to T0 spaces of continuous posets with their Scott topology. The main results are: (1) A T0 space (X,τ) is a C-space iff the S-convergence coincides with convergence in the topology τ iff (X,τ) is a web space and the liminf convergence coincides with convergence in the supremum of τ and the lower topology; (2) A T0 space is a WC-space iff the liminf convergence is topological for the liminf topology.