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# 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: Bull. 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: Fundamenta Mathematicae, Volume 34.)
- 3 E. S. Pondiczery, Power problems in abstract spaces, Duke Math. J. 11 (1944), 835–837.

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Dense, Separable

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## Mathematics Subject Classification

54D65*no label found*54A25

*no label found*

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