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.

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.

{{cite book | title=Encyclopedia of Distances | first1=Michel Marie | last1=Deza | first2=Elena | last2=Deza|author2-link=Elena Deza | publisher=Springer-Verlag | year=2012 | isbn=978-3642309588 | page=64 }}
{{cite book | title=Encyclopedia of General Topology | url=https://archive.org/details/encyclopediagene00hart | url-access=limited | first1=K.P. | last1=Hart |first2=Jun-iti | last2=Nagata | first3=J.E. | last3=Vaughan | publisher=Elsevier | year=2003 | isbn=0080530869 | page=[https://archive.org/details/encyclopediagene00hart/page/n263 273] }}

[https://web.archive.org/web/20080509123602/http://www1.elsevier.com/homepage/sac/opit/book.pdf A book of unsolved problems in topology; see page 91 for cosmic spaces]