discrete topology of cardinality 6

Theorem 1.15 shows that for any cardinal number m and any metrizable space X seven conditions are equivalent.
Included in these seven are the following:

(a) every (closed) discrete subspace of X has cardinality less than or equal to m;
(b) X has a dense subset of cardinality less than or equal to m.

These conditions may be formulated in an arbitrary topological space and we consider how they are related in such a wider setting.

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