When do the Upper Kuratowski Topology (Homeomorphically, Scott Topology) and the Co-compact Topology Coincide?

Abstract

Abstract.A topology is called consonant if the corresponding upper Kuratowski topology on closed sets coincides with the co-compact topology, equivalently if each Scott open set is compactly generated. It is proved that Cechcomplete topologies are consonant and that consonance is not preserved by passage to G^-sets, quotient maps and finite products. However, in the class of the regular spaces, the product of a consonant topology and of a locally compact topology is consonant. The latter fact enables us to characterize the topologies generated by some T-convergences.

Products and projective limits of continuous valuations on T0 spaces

“…Hence, we define the following. Here is the way-below relation on R + ; we have r s if and only if r = 0 or r < s. Every locally compact space is consonant, and the Dolecki-Greco-Lechicki theorem states that all regularČech-complete spaces are consonant (Dolecki et al, 1995). The latter include all complete metric spaces, even not locally compact, in their open ball topology.…”

The quasi Scott (Lawson) topology and q-continuous (q-algebraic) complete lattices

“…Many researchers are interested in continuous (algebraic) lattices, Scott (Lawson) topology, and their applications (see, for example, [1,4,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]). In section 2 we define and study the quasi Scott topology on a complete lattice.…”

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].…”