## On $$d$$-separability of powers and $$C_p(X)$$.(English)Zbl 1134.54002

A space is $$d$$-separable if it has a dense set representable as a countable union of discrete subsets. Let $$\hat s(X)$$ be the smallest cardinal $$\lambda$$ such that $$X$$ has no discrete subspace of size $$\lambda$$. If $$X$$ is T$$_1$$ and $$\kappa$$ is a cardinal with $$\hat s(X^\kappa)>d(X)$$ then $$X^\kappa$$ is $$d$$-separable, while if $$X$$ is compact T$$_2$$ then $$\hat s(X^2)>d(X)$$. Consequently if $$X$$ is T$$_1$$ then $$X^{d(X)}$$ is $$d$$-separable while if $$X$$ is compact T$$_2$$ then $$X^\omega$$ is $$d$$-separable. Examples are also given of spaces which are not $$d$$-separable.

### MSC:

 54B10 Product spaces in general topology 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
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### References:

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