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.

External links

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