Convergence-theoretic mechanisms behind product theorems

Core compactness and diagonality in spaces of open sets

“…While this proposition easily follows from general results in [7] or [20], and Galois connections, we provide a self-contained proof.…”

“…Since P is an ideal subbase, there is P ∈ P ∩ A. In order to investigate when T [X, $] is core compact, we will need notions and results from [7]. The concrete endofunctor Epi T of the category of convergence spaces (and continuous maps) is defined (on objects) by…”

“…While the latest is well-known, and follows for instance from the results of [7], it seems difficult to find an elementary argument in the literature, which is why we include the following proposition, which also illustrates the usefulness of Lemma 5.2. Proof.…”

“…In [7], it is shown that a convergence space X is T -core compact if and only if [X, $] is pretopological. Therefore, if X is topological, [X, $] is topological whenever it is pretopological.…”

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

“…In this survey paper we not only present a collection of our recent results related to hyperconvergence [11], [17], [16], [15], [14], but also refine several of them and clarify certain links between them. Therefore we have tried to make this paper self-contained providing full proofs in most cases.…”