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 "T"1 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.
Unsolved problems.
It is unknown as to whether "X" is cosmic if:
a) "X"2 contains no uncountable discrete space;
b) the countable product of "X" with itself is hereditarily separable and hereditarily Lindelöf.