A generalization of the Sierpiński theorem

Abstract:

Sierpiński’s theorem admits the following generalization. Let $n$ be a nonnegative integer and $X$ a compact Hausdorff space. If $\left \{ {{F_i}\left | {i \in {\mathbf {N}}} \right .} \right \}$ is a countable closed covering of $X$ such that $\left ( {{F_i} \cap {F_j}} \right ) < n$ for distinct $i$ and $j$ in ${\mathbf {N}}$, then every continuous mapping from ${F_1}$ into the $n$-sphere ${S^n}$ is extendable over $X$.