Perfect maps between submetrizable spaces

TL;DR

This paper explores whether perfect maps between paracompact submetrizable spaces can be metrized to preserve their perfectness under induced metrics.

Contribution

It addresses a question about the metric submetrization of perfect maps between paracompact submetrizable spaces.

Findings

The paper provides conditions under which perfect maps can be metrized.

It establishes a link between perfect maps and submetrizability.

The results clarify the structure of submetrizable spaces under perfect maps.

Abstract

We investigate a question posed by Huaipeng Chen: if $X$ and $Y$ are paracompact submetrizable spaces and $f:X\to Y$ is a perfect map, can $X$ and $Y$ be submetrized by metrics $\rho$ and $d$ respectively such that $f$ remains perfect with respect to the induced topologies?