:: : I am going to stop now in case I get too carried away; ) but these problems illustrate the diversity of general topology and how simple ideas such as the OP's question may lead to such interesting questions ( for instance, in question one, we know that every separable space satisfies the countable chain condition; the obvious question is whether there exists a non-separable space which satisfies the countable chain condition ( that is, does the converse hold ? ).
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