Continuous selections, prime number and a covering type property

TL;DR

This paper investigates conditions under which a Hausdorff space admits continuous selections for finite subsets, revealing a connection between prime and composite subset sizes and introducing a covering property characterization.

Contribution

It establishes that continuous selections over small subsets imply selections over larger non-prime subset sizes, and characterizes these properties via a covering-type condition.

Findings

Spaces with continuous selections over small subsets extend to certain larger subsets.

Continuous selections over all finite subsets are equivalent to those over prime-sized subsets.

A covering property characterizes the existence of continuous selections over larger subsets.

Abstract

Let $(X,\tau)$ be a Hausdorff space and $n\in\omega$. We prove that if $X$ admits a continuous selection over $\mathcal{F}_{n}(X)$ (nonempty subsets of $X$ of cardinality at most $n$), then for every $n\leq m\leq 2n$ such that $m$ is not a prime number, $X$ admits a continuous selection over $[X]^m$ (subsets of $X$ of cardinality $m$). As a consequence of this, a space $X$ admits a continuous selection for every natural number if and only if the same is true for every prime number. For Hausdorff spaces $(X,\tau)$ which admit continuous selections over $[X]^2$, we characterize the existence of continuous selections over $[X]^n$ for $n\geq 2$, in terms of a covering-type property.