✍ Scribed by Ahmed Bouziad
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A space X is said to be consonant if, on the set of closed subsets of X, the upper Kuratowski topology coincides with the co-compact topology. It is known that tech-complete spaces are consonant and that consonance is neither preserved by Gg subsets nor stable under products. We show that all Gg subspaces of a consonant space X are consonant if the Vietoris topology on compact subsets of X is hereditarily Baire; and that is always the case if all compact subspaces of X are separable and of countable character in X. Spaces which are G6 subspaces of consonant paracompact p-spaces are also shown to be consonant. Concerning products, we show that the product of a consonant paracompact p-space and a tech-complete space is consonant. We also answer some questions of Nogura and Shakhmatov related to product and topological sum operations in the class of regular consonant spaces. 0 1998 Elsevier Science B.V.
📜 SIMILAR VOLUMES
In this note we describe the Gg-closure of Zx in 26x "Ising this description, we obtain the following theorem: 2x is1 Gg-closed in 2px if and on F' X is LiudeEf. An immcdiatc consequence of this result is the fact that 2x is reakomp if X is Lindelijf. Concerning the realcompactness of 2x, we show th