# Cosmic space

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In mathematics, particularly topology, a **cosmic space** is any topological space that is a continuous image of some separable metric space. Equivalently (for regular *T*_{1} spaces but not in general), a space is cosmic if and only if it has a countable network; namely a countable collection of subsets of the space such that any open set is the union of a subcollection of these sets.

Cosmic spaces have several interesting properties. There are a number of unsolved problems about them.

## Examples and properties[edit]

- Any open subset of a cosmic space is cosmic since open subsets of separable spaces are separable.
- Separable metric spaces are trivially cosmic.

## Unsolved problems[edit]

It is unknown as to whether *X* is cosmic if:

a) *X*^{2} contains no uncountable discrete space;

b) the countable product of *X* with itself is hereditarily separable and hereditarily Lindelöf.

## References[edit]

- Deza, Michel Marie; Deza, Elena (2012).
*Encyclopedia of Distances*. Springer-Verlag. p. 64. ISBN 3642309585. - Hart, K.P.; Nagata, Jun-iti; Vaughan, J.E. (2003).
*Encyclopedia of General Topology*. Elsevier. p. 273. ISBN 0080530869.