Continuous selections, prime number and a covering type property

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 $n\le m\le 2n$ 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 $n\ge 2$, in terms of a covering-type property.