Some topological properties of spaces of Lipschitz continuous maps on quasi-metric spaces

摘要

Given any quasi-metric space X, d, we can form the space LX of all lower semicontinuous maps from X to R+, where X is given the d-Scott topology. We give LX the Scott topology of the pointwise ordering. We can then form the subspace L-alpha(X, d) of alpha-Lipschitz continuous maps from X, d to R+ (alpha is an element of R+). We show that, when X, d is continuous Yoneda-complete, L-alpha (X, d) is stably compact, and that its topology coincides with the compact-open topology and with the topology of pointwise convergence. We also show that the space L-infinity(X, d) of all Lipschitz continuous maps from X, d to R+ has a topology that is determined by the topologies of L-alpha(X,d), alpha 0, if X, d is Lipschitz regular. (C) 2020 Elsevier B.V. All rights reserved.