We show that if a Hausdorff topological space satisfies one of the following properties:
a) has a countable, discrete dense subset and is hereditarily collectionwise Hausdorff;
b) has a discrete dense subset and admits a countable base;
then the existence of a (continuous) weak selection on implies weak orderability. As a special case of either item a) or b), we obtain the result for every separable metrizable space with a discrete dense subset.
Assuming Martin’s Axiom, we provide an example of two Fréchet-Urysohn -spaces, whose product is a non-Fréchet-Urysohn -space. This gives a consistent negative answer to a question raised by T. Nogura.
A topological space is KC when every compact set is closed and SC when every convergent sequence together with its limit is closed. We present a complete description of KC-closed, SC-closed and SC minimal spaces. We also discuss the behaviour of the finite derived set property in these classes.
An estimate for the Novak number of a hyperspace with the Vietoris topology is given. As a consequence it is shown that this cardinal function can decrease passing from a space to its hyperspace.
We calculate the density of the hyperspace of a metric space, endowed with the Hausdorff or the locally finite topology. To this end, we introduce suitable generalizations of the notions of totally bounded and compact metric space.
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