TOPOLOGICAL SPACES GENERATED BY DISCRETE SUBSPACES

In [3] the authors initiated a systematic study of the property of a space to be generated by its discrete subsets. Discretely generated properties seems to be interesting in themselves due to their good categorical behaviour: discrete generability is hereditary; each compact space of countable tightness is discretely generated; FrechetUrysohn is discretely generated. Such a property turns out to be not only interesting, but also the base for many nice questions. Discrete generability has surprising relationships with classical properties. According to the article [1] sequential spaces is generated by discrete sets, but in this work we will show an example which sequential space is not discrete generated space. A lot of properties are preserved by making product. The product of Hausdorff spaces is a Hausdorff space, the product of compacts is a compact, the countable product of metrizable spaces is metrizable. However it is not case of the discrete generability. We will show a simple counter-example. Additionally, the product of Frechet-Urysohn spaces are not discretely generated space was shown by example.