Injections into Function Spaces over Compacta

TL;DR

This paper investigates the topological properties of function spaces over compacta, especially focusing on injective maps from spaces over GO-spaces and constructing specific continuous bijections with group isomorphism properties.

Contribution

It establishes conditions under which the Dedekind remainder of a GO-space is hereditarily paracompact and constructs explicit continuous bijections with group isomorphism between certain function spaces.

Findings

Injective maps from Cp over GO-spaces to Cp over compacta imply hereditarily paracompact Dedekind remainders.

Constructed continuous bijections between Cp(tau, {0,1}) and subgroups of Cp(tau+1, {0,1}) for uncountable cofinality ordinals.

Demonstrated group isomorphisms in the context of function spaces over specific ordinal spaces.

Abstract

We study the topology of X given that Cp(X) injects into Cp(Y), where Y is compact. We first show that if Cp over a GO-space (="subspace of a lineraly ordered space") injects into Cp over a compactum, then the Dedekind remainder of the GO-space is hereditarily paracompact. Also, for each ordinal tau of uncountable cofinality, we construct a continuous bijection of Cp(tau, {0,1}) onto a subgroup of Cp(tau+1, {0,1}), which is in addition a group isomorphism.