Applied General Topology - Vol 03, No 2 (2002)https://riunet.upv.es:443/handle/10251/820012024-06-19T20:43:08Z2024-06-19T20:43:08ZFinite approximation of stably compact spacesSmyth, M.B.Webster, J.https://riunet.upv.es:443/handle/10251/820822023-11-21T11:47:17Z2017-05-31T09:06:49ZFinite approximation of stably compact spaces
Smyth, M.B.; Webster, J.
[EN] Finite approximation of spaces by inverse sequences of graphs (in the category of so-called topological graphs) was introduced by Smyth, and developed further. The idea was subsequently taken up by Kopperman and Wilson, who developed their own purely topological approach using inverse spectra of finite T0-spaces in the category of stably compact spaces. Both approaches are, however, restricted to the approximation of (compact) Hausdorff spaces and therefore cannot accommodate, for example, the upper space and (multi-) function space constructions. We present a new method of finite approximation of stably compact spaces using finite stably compact graphs, which when the topology is discrete are simply finite directed graphs. As an extended example, illustrating the problems involved, we study (ordered spaces and) arcs.
2017-05-31T09:06:49ZCofinitely and co-countably projective spacesMendoza Iturralde, PabloTkachuk, Vladimir V.https://riunet.upv.es:443/handle/10251/820812023-11-21T11:47:17Z2017-05-31T09:00:59ZCofinitely and co-countably projective spaces
Mendoza Iturralde, Pablo; Tkachuk, Vladimir V.
[EN] We show that X is cofinitely projective if and only if it is a finite union of Alexandroff compactatifications of discrete spaces. We also prove that X is co-countably projective if and only if X admits no disjoint infinite family of uncountable cozero sets. It is shown that a paracompact space X is co-countably projective if and only if there exists a finite set B C X such that B C U ϵ τ (X) implies │X\U│ ≤ ω. In case of existence of such a B we will say that X is concentrated around B. We prove that there exists a space Y which is co-countably projective while there is no finite set B C Y around which Y is concentrated. We show that any metrizable co-countably projective space is countable. An important corollary is that every co-countably projective topological group is countable.
2017-05-31T09:00:59ZExtendible spacesSchellekens, M.P.https://riunet.upv.es:443/handle/10251/820802023-11-21T11:47:17Z2017-05-31T08:58:18ZExtendible spaces
Schellekens, M.P.
[EN] The domain theoretic notion of lifting allows one to extend a partial order in a trivial way by a minimum. In the context of Quantitative Domain Theory partial orders are represented as quasi-metric spaces. For such spaces, the notion of the extension by an extremal element turns out to be non trivial.
To some extent motivated by these considerations, we characterize the directed quasi-metric spaces extendible by an extremum. The class is shown to include the S-completable directef quasi-metric spaces. As an application of this result, we show that for the case of the invariant quasi-metric (semi)lattices, weightedness can be characterized by order convexity with the extension property.
2017-05-31T08:58:18ZOn the structure of completely useful topologiesBosi, GianniHerden, Gerhardhttps://riunet.upv.es:443/handle/10251/820782023-11-21T11:47:17Z2017-05-31T08:44:12ZOn the structure of completely useful topologies
Bosi, Gianni; Herden, Gerhard
[EN] Let X be an arbitrary set. Then a topology t on X is completely useful if every upper semicontinuous linear preorder on X can be represented by an upper semicontinuous order preserving real-valued function. In this paper we characterize in ZFC (Zermelo-Fraenkel + Axiom of Choice) and ZFC+SH (ZFC + Souslin Hypothesis) completely useful topologies on X. This means, in the terminology of mathematical utility theory, that we clarify the topological structure of any type of semicontinuous utility representation problem.
2017-05-31T08:44:12ZLarge and small sets with respect to homomorphisms and products of groupsGusso, Riccardohttps://riunet.upv.es:443/handle/10251/820772023-11-21T11:47:17Z2017-05-31T08:42:12ZLarge and small sets with respect to homomorphisms and products of groups
Gusso, Riccardo
[EN] We study the behaviour of large, small and medium subsets with respect to homomorphisms and products of groups. Then we introduce the definition af a P-small set in abelian groups and we investigate the relations between this kind of smallness and the previous one, giving some examples that distinguish them.
2017-05-31T08:42:12ZStrengthening connected Tychonoff topologiesShakhmatov, DimitriTkachenko, MikhailTkachuk, Vladimir V.Wilson, Richard G.https://riunet.upv.es:443/handle/10251/820602023-11-21T11:47:17Z2017-05-31T07:31:17ZStrengthening connected Tychonoff topologies
Shakhmatov, Dimitri; Tkachenko, Mikhail; Tkachuk, Vladimir V.; Wilson, Richard G.
[EN] The problem of whether a given connected Tychonoff space admits a strictly finer connected Tychonoff topology is considered. We show that every Tychonoff space X satisfying ω (X) ≤ c and c (X) ≤ N0 admits a finer strongly σ-discrete connected Tychonoff topology of weight 2c. We also prove that every connected Tychonoff space is an open continuous image of a connected strongly σ-discrete submetrizable Tychonoff space. The latter result is applied to represent every connected topological group as a quotient of a connected strongly σ-discrete submetrizable topological group
2017-05-31T07:31:17ZA curious example involving ordered compactificationsRichmond, Thomas A.https://riunet.upv.es:443/handle/10251/820592023-11-21T11:47:17Z2017-05-31T07:28:41ZA curious example involving ordered compactifications
Richmond, Thomas A.
[EN] For a certain product X x Y where X is compact, connected, totally ordered space, we find that the semilattice K0 (X x Y) of ordered compactifications of X x Y is isomorphic to a collection of Galois connections and to a collection of functions F which determines a quasi-uniformity on an extended set X U {+∞}, from which the topology and order on X is easily recovered. It is well-known that each ordered compactification of an ordered space X x Y corresponds to a totally bounded quasi-uniformity on X x Y compatible with the topology and order on X x Y, and thus K0 (X x Y) may be viewed as a collection of quasi-uniformities on X x Y. By the results here, these quasi-uniformities on X x Y determine a quasi-uniformity on the related space X U {+∞}.
2017-05-31T07:28:41Z