[Paper Review] Core-compactness of Smyth powerspaces

Zhenchao Lyu, Xiaodong Jia|arXiv (Cornell University)|Jul 10, 2019

Advanced Topology and Set Theory8 references4 citations

TL;DR

This paper establishes that the Smyth powerspace $Q(X)$ of a topological space $X$ is core-compact if and only if $X$ is locally compact. The authors prove that core-compactness is not preserved by the Smyth powerspace construction in general, as there exist core-compact spaces that are not locally compact, and their Smyth powerspaces fail to be core-compact.

ABSTRACT

We prove that the Smyth powerspace Q(X) of a topological space X is core-compact if and only if X is locally compact. As a straightforward consequence we obtain that the Smyth powerspace construction does not preserve core-compactness generally.

Motivation & Objective

• To investigate whether core-compactness is preserved under the Smyth powerspace construction.
• To clarify the relationship between core-compactness of $Q(X)$ and local compactness of $X$.
• To demonstrate that core-compactness is not preserved in general by constructing a counterexample.
• To establish a characterization of core-compact Smyth powerspaces via local compactness of the base space.

Proposed method

• Prove that $Q(X)$ is a c-space if and only if $X$ is locally compact, using the upper Vietoris topology on compact saturated sets.
• Use the equivalence between c-spaces and prime-continuous open set lattices ($O(X)$) to analyze core-compactness.
• Apply the basis criterion for prime-continuity in continuous lattices, verifying the condition $\Box U \subseteq \Box V \cup \Box W$ implies $\Box U \subseteq \Box V$ or $\Box U \subseteq \Box W$.
• Leverage known results: $X$ is a c-space iff $O(X)$ is prime-continuous, and there exist core-compact spaces that are not locally compact.
• Use contradiction to show that if $Q(X)$ is core-compact, then $O(Q(X))$ is prime-continuous, leading to the conclusion that $X$ must be locally compact.
• Apply the result from [9] that there exists a core-compact space not locally compact, to show $Q(X)$ fails to be core-compact in such cases.

Experimental results

Research questions

• RQ1 Under what conditions is the Smyth powerspace $Q(X)$ core-compact?
• RQ2 Does the Smyth powerspace construction preserve core-compactness for all topological spaces?
• RQ3 What is the relationship between local compactness of $X$ and core-compactness of $Q(X)$?
• RQ4 Can a core-compact space $X$ have a Smyth powerspace $Q(X)$ that is not core-compact?
• RQ5 Is there a characterization of core-compact Smyth powerspaces in terms of properties of the base space $X$?

Key findings

• The Smyth powerspace $Q(X)$ is core-compact if and only if $X$ is locally compact.
• The Smyth powerspace construction does not preserve core-compactness in general, as shown by counterexamples.
• Core-compactness of $Q(X)$ implies that $O(Q(X))$ is prime-continuous, which leads to the conclusion that $X$ must be locally compact.
• The equivalence between $Q(X)$ being a c-space and $X$ being locally compact is established via the upper Vietoris topology and basis arguments.
• The existence of a core-compact space that is not locally compact (from [9]) implies that its Smyth powerspace is not core-compact.
• The proof relies on the fact that $\{\Box U \mid U \in O(X)\}$ forms a basis for $O(Q(X))$, and the prime-continuity condition holds only when $X$ is locally compact.