A generalization of the Sierpiński theorem
Jan J. Dijkstra · Proceedings of the American Mathematical Society · 1984
Sierpiński’s theorem admits the following generalization. Let n n be a nonnegative integer and X X a compact Hausdorff space. If { F i | i ∈ N } \left \{ {{F_i}\left | {i \in {\mathbf {N}}} \right .} \right \} is a countable closed covering of X X such that ( F i ∩ F j ) > n \left ( {{F_i} \cap {F_j}} \right ) > n for distinct i i and j j in N {\mathbf {N}} , then every continuous mapping from F 1 {F_1} into the n n -sphere S n {S^n} is extendable over X X .