Hewitt-Marczewski-Pondiczery theorem

The Hewitt–Marczewski–Pondiczery Theoremis a result on the density of products of topological spaces.This theorem was arrived at independently byHewitt[1], Marczewski[2] and Pondiczery[3]in the 1940s.

Theorem.

Let $\kappa$ be an infinite cardinal number and $S$ an index setof cardinality at most ${\mathrm{2}}^{\kappa }$.If $X_s$ $\mathrm{(}s\mathrm{\in }S\mathrm{)}$ are topological spaces with $d\mathit{}\mathrm{(}{X}_s\mathrm{)}\mathrm{\le }\kappa$ then

$$d\left(\prod _{s\in S}{X}_s\right)\le \kappa .$$

The special case $\kappa ={\mathrm{\aleph }}_0$says that the product of at most continuum many separable spaces is separable.