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

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

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