Cosmic space

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.

Examples and properties

• 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

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.

References

• 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. https://archive.org/details/encyclopediagene00hart. 

External links

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