$Cp(X)$ for Hattori Spaces

TL;DR

This paper investigates various topological properties of Hattori spaces, providing complete characterizations for some and partial results for others, especially concerning compactness, metrizability, semi-stratifiability, and properties of function spaces.

Contribution

It offers new criteria for compactness and metrizability in Hattori spaces and establishes conditions under which function spaces are normal or Lindelöf, advancing understanding of these spaces.

Findings

Complete characterization of compact sets in Hattori spaces.

Conditions for Hattori spaces to be quasi-metrizable and semi-stratifiable.

Proved that if the complement of A is analytic, then Cp(H(A)) is not normal.

Abstract

Motivated by the main results of the articles by Hattori and Bouziad, we seek to answer the following questions about Hattori spaces. Let A be a subset of the real line, then: Given a compact set $K$ in the Euclidean topology, under what conditions is $K$ compact in the Hattori space $H(A)$? When is $H(A)$ a quasi-metrizable space? When is $H(A)$ a semi-stratifiable space? When is $C_p(H(A))$ a normal space? When is $C_p(H(A))$ a Lindel\"of space? We obtain complete answers for 3 out of these 5 questions, while the last ones remain with partial answers, among them: \ Theorem: If $\mathbb{R}\setminus A$ is analytic, then $C_p(H(A))$ is not normal. Moreover when we work on the Solovay Model we can improve the previous result to only require $\mathbb{R}\setminus A$ to be uncountable.