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

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

Topology
Topological spaces