# 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 ${\mathrm{2}}^{\kappa}$.
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.

## 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 |
---|---|

Canonical name | HewittMarczewskiPondiczeryTheorem |

Date of creation | 2013-03-22 17:16:56 |

Last modified on | 2013-03-22 17:16:56 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 6 |

Author | yark (2760) |

Entry type | Theorem |

Classification | msc 54D65 |

Classification | msc 54A25 |

Related topic | Dense |

Related topic | Separable |