- Cosmic space
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In mathematics, particularly topology, cosmic spaces have several interesting properties. There are a number of unsolved problems about them.
Contents
Formal definition
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.
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 Lindelof.
References
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