For a topological space X, ℒ(X) is the set of lower semicontinuous maps from X to the extended non-negative real numbers. Let me give it the pointwise ordering, and the Scott topology of that ordering.
ℒ(X) is also a semitopological cone, in the sense of Klaus Keimel [1]. There is a pointwise addition operation +, an operation of multiplication by non-negative real numbers, which satisfy the expected algebraic properties: + is associative, commutative, has the zero function as neutral element, scalar multiplication by a non-negative real number a distributes over addition, etc. Additionally, the scalar multiplication operation and the addition operation are separately continuous. By the Banaschewski-Lawson-Ershov observation, scalar multiplication is even jointly continuous.
I am regularly encountering situations where I need to assume that ℒ(X) is locally convex. A semitopological cone is locally convex if and only if every point has a base of convex open neighborhoods. A convex set is a set A such that for all x, y ∈ A, for every a ∈ [0, 1], ax+(1–a)y ∈ A. Local convexity means that for every point x and every open neighborhood U of x, there is a convex open neighborhood V of x that is included in U.
Question 1: Under what conditions on X is ℒ(X) a locally convex semitopological cone?
A partial answer can be found in [2]. Lemma 3.6 of [2] states that every ⊙-consonant space X is such that ℒ(X) is locally convex. A space is ⊙-consonant if and only if any finite copower of X is consonant, and I have already mentioned this property in the November 2024 post. Every core-compact space X, every LCS-complete space X is ⊙-consonant.
But I don’t even know whether there is single example of a space X where ℒ(X) is not locally convex.
The proof of Lemma 3.6 of [2] relies on the fact that, if X is ⊙-consonant, then the Scott topology on ℒ(X) coincides with the compact-open topology; and it is easy to see that with that second topology, every h ∈ ℒ(X) has a base of convex open neighborhoods.
A related question is as follows. Above, I have said that scalar multiplication in a semitopological cone is always jointly continuous. Addition is only separately continuous in general. A semitopological cone in which addition is jointly continuous is called a topological cone.
Question 2: Under what conditions on X is ℒ(X) a topological cone?
I know that ℒ(X) is a topological cone when X is core-compact. In that case, ℒ(X) is even a continuous d-cone, namely a semitopological cone that is also a continuous dcpo [1, Definition 6.5]; and every continuous d-cone is topological (as well as being sober, locally convex, locally convex-compact, among other properties).
As with local convexity, I don’t even know whether there is single example of a space X where ℒ(X) is not topological.
- Klaus Keimel. Topological cones: functional analysis in a T0-setting. Semigroup forum 77(1), pages 109-142, 2008.
- Matthew de Brecht, Jean Goubault-Larrecq, Xiaodong Jia, Zhenchao Lyu. Domain-complete and LCS-complete spaces. Proceedings of the 8th ISDT, Electronic Notes in Theoretical Computer Science 345:3–35, 2019.
— Jean Goubault-Larrecq (March 28th, 2025)
