# Hewitt-Marczewski-Pondiczery theorem

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

###### Theorem.

Let $\kappa$ be an infinite cardinal number and $S$ an index set of cardinality at most $2^{\kappa}$. If $X_{s}$ $(s\in S)$ are topological spaces with $d(X_{s})\leq\kappa$ then

 $d\left(\prod_{s\in S}X_{s}\right)\leq\kappa.$

The special case $\kappa=\aleph_{0}$ says that the product of at most continuum many separable spaces is separable.

## References

• 1 Edwin Hewitt, A remark on density characters, Bull. Amer. Math. Soc. 52 (1946), 641–643. (This paper is available as a PDF file from the AMS website: http://www.ams.org/journals/bull/1946-52-08/home.htmlBull. Amer. Math. Soc., Volume 52, Number 8.)
• 2 Edward Marczewski, Séparabilité et multiplication cartésienne des espaces topologiques, Fund. Math. 34 (1947), 127–143. (This paper is available as a PDF file from the Polish Virtual Library of Science: http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=34&jez=enFundamenta Mathematicae, Volume 34.)
• 3 E. S. Pondiczery, Power problems in abstract spaces, Duke Math. J. 11 (1944), 835–837.
Title Hewitt-Marczewski-Pondiczery theorem HewittMarczewskiPondiczeryTheorem 2013-03-22 17:16:56 2013-03-22 17:16:56 yark (2760) yark (2760) 6 yark (2760) Theorem msc 54D65 msc 54A25 Dense Separable