On the coincidence of the upper Kuratowski topology with the cocompact topology

Abstract

The hyperconvergence (upper Kuratowski convergence) is the coarsest convergence on the set of closed subsets of a convergence space that makes the canonical evaluation continuous. Sundry reflective and coreflective properties of hyperconvergences are characterized in terms of the underlying convergence.

Equilibrium problems techniques in the qualitative analysis of quasi-hemivariational inequalities

“…Every metric space and more generally, every Fréchet-Urysohn space is a sequential space. A space X is called Fréchet-Urysohn space if whenever x is in the closure of a subset B of X , there exists a sequence in B converging to x, see Engelking [10], Alleche and Calbrix [11] for further details.…”

Semicontinuity of bifunctions and applications to regularization methods for equilibrium problems

“…Our restriction to R n is motivated by the fact that the proof of Lemma 1 is based on the Fréchet-Urysohn property which is of course verified by R n . A space verifies the Fréchet-Urysohn property if a point is in the closure of a subset if and only if it is a limit of a sequence in the subset (see [4,13]). …”

Vietoris topology on partial maps with compact domains

“…Our next theorem gives a sufficient condition for hereditary Baireness of (P K (X, Y ), τ V ). Recall, that X is consonant [15,16], provided the upper Kuratowski topology and the cocompact topology coincide on the hyperspace of closed subsets of X ;Čech-complete spaces are consonant [16], but there are separable metrizable hereditarily Baire non-consonant spaces [1].…”