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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) X contains no uncountable discrete space;
b) the countable product of X with itself is hereditarily separable and hereditarily Lindelöf.
References
- Deza, Michel Marie; Deza, Elena (2012). Encyclopedia of Distances. Springer-Verlag. p. 64. ISBN 3642309585.
- Hart, K.P.; Nagata, Jun-iti; Vaughan, J.E. (2003). Encyclopedia of General Topology. Elsevier. p. 273. ISBN 0080530869.