Cosmic space

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (June 2022) (Learn how and when to remove this message)

In mathematics, particularly topology, a cosmic space is any topological space that is a continuous image of some separable metric space. Equivalently (for regular T₁ 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.

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

It is unknown as to whether X is cosmic if:

a) X² contains no uncountable discrete space;

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

• Deza, Michel Marie; Deza, Elena (2012). Encyclopedia of Distances. Springer-Verlag. p. 64. ISBN 978-3642309588.
• Hart, K.P.; Nagata, Jun-iti; Vaughan, J.E. (2003). Encyclopedia of General Topology. Elsevier. p. 273. ISBN 0080530869.

• A book of unsolved problems in topology; see page 91 for cosmic spaces