# A compact $G_\delta$ of a polyadic spaces.

For any locally compact topological space $X$, the (Alexandroff) one-point compactification of $X$ is obtained by adding one extra point $\infty$ and defining the topology on $X \cup \{\infty\}$ to consist of the open sets of $X$ together with the sets of the form $U \cup \{\infty\}$, where $U$ is an open subset of $X$ and $X \backslash U$ is compact. In 1970 Polyadic spaces were defined by Mrowka as a Hausdorff space $X$ which is a continuous image of some power of the one-point compactification of a discrete space $\kappa$, denoted by $\alpha\kappa^\tau$. Gerlits proved in his paper entitle 'Generalization of Dyadicity' that a compact $G_\delta$ of a polyadic space is polyadic. In his proof he assumed that any open set is a union of finitely many base sets. Why did he assume that?

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