Spaces l-Dominated by I or R

TL;DR

This paper investigates the conditions under which the unit interval and the real line can $ ext{l}$-dominate certain compact metrizable spaces, revealing characterizations based on fd-height and metrizability.

Contribution

It provides new characterizations of spaces $ ext{l}$-dominated by the unit interval and reals, linking $ ext{l}$-domination to properties like fd-height and metrizability.

Findings

The unit interval $ ext{l}$-dominates compact metrizable spaces with finite fd-height.

Spaces $ ext{l}$-dominated by the unit interval are necessarily compact, metrizable, and have countable fd-height.

Similar results are established for spaces $ ext{l}$-dominated by the reals.

Abstract

If $X$ is compact metrizable and has finite fd-height then the unit interval, $I$, $\ell$-dominates $X$, in other words, there is a continuous linear map of $C_p(I)$ onto $C_p(X)$. If the unit interval $\ell$-dominates a space $X$ then $X$ is compact metrizable and has countable fd-height. Similar results are given for spaces $\ell$-dominated by the reals.