Mazur-like topological linear spaces and their products

summary:Topological linear spaces having the property that some sequentially continuous linear maps on them are continuous, are investigated. It is shown that such properties (and close ones, e.g., bornological-like properties) are closed under large products

Characters of ultrafilters and tightness of products of fans

AbstractAs applications of productivity of coreflective classes of topological spaces, the following results will be proved: (1) Characters of points of βN∖N are not smaller than any submeasurable cardinal less or equal to 2ω. (2) If κ is a submeasurable cardinal and S is a sequential fan with κ many spines then the tightness of the κ-power of S is equal to κ. In fact, a little more general results are proved

Normality in terms of distances and contractions

The main purpose of this paper is to explore normality in terms of distances between points and sets. We prove some important consequences on realvalued contractions, i.e. functions not enlarging the distance, showing that as in the classical context of closures and continuous maps, normality in terms of distances based on an appropriate numerical notion of $\gamma$-separation of sets, has far reaching consequences on real valued contractive maps, where the real line is endowed with the Euclidean metric. We show that normality is equivalent to (1) separation of $\gamma$-separated sets by some Urysohn contractive map, (2) to Kat\v{e}tov-Tong's interpolation, stating that for bounded positive realvalued functions, between an upper and a larger lower regular function, there exists a contractive interpolating map and (3) to Tietze's extension theorem stating that certain contractions defined on a subspace can be contractively extended to the whole space. The appropriate setting for these investigations is the category of approach spaces, but the results have (quasi)-metric counterparts in terms of non-expansive maps. Moreover when restricted to topological spaces, classical normality and its equivalence to separation by a Urysohn continuous map, to Kat\v{e}tov-Tong's interpolation for semicontinuous maps and to Tietze's extension theorem for continuous maps are recovered

On subsequential spaces

AbstractSimple generators for the coreflective category of subsequential spaces, one of them countable, are constructed. Every such must have subsequential order ω1. Subsequentialness is a local property and a countable property, both in a strong sense. A T2-subsequential space may be pseudocompact without being sequential, in contrast to T2-subsequential compact (countably compact, sequentially compact) spaces all being sequential. A compact subsequential space need not be sequential

On classes of T0 spaces admitting completions

[EN] For a given class X of T0 spaces the existence of a subclass C, having the same properties that the class of complete metric spaces has in the class of all metric spaces and non-expansive maps, is investigated. A positive example is the class of all T0 spaces, with C the class of sober T0 spaces, and a negative example is the class of Tychonoff spaces. We prove that X has the previous property (i.e., admits completions) whenever it is the class of T0 spaces of an hereditary coreflective subcategory of a suitable supercategory of the category Top of topological spaces. Two classes of examples are provided.Giuli, E. (2003). On classes of T0 spaces admitting completions. Applied General Topology. 4(1):143-155. doi:10.4995/agt.2003.2016.SWORD1431554

Comparing Cartesian-closed Categories of (Core) Compactly Generated Spaces

It is well known that, although the category of topological spaces is not cartesian closed, it possesses many cartesian closed full subcategories, e.g.: (i) compactly generated Hausdorff spaces; (ii) quotients of locally compact Hausdorff spaces, which form a larger category; (iii) quotients of locally compact spaces without separation axiom, which form an even larger one; (iv) quotients of core compact spaces, which is at least as large as the previous; (v) sequential spaces, which are strictly included in (ii); and (vi) quotients of countably based spaces, which are strictly included in the category (v). We give a simple and uniform proof of cartesian closedness for many categories of topological spaces, including (ii)–(v), and implicitly (i), and we also give a self-contained proof that (vi) is cartesian closed. Our main aim, however, is to compare the categories (i)–(vi), and others like them. When restricted to Hausdorff spaces, (ii)–(iv) collapse to (i), and most non-Hausdorff spaces of interest, such as those which occur in domain theory, are already in (ii). Regarding the cartesian closed structure, finite products coincide in (i)–(vi). Function spaces are characterized as coreflections of both the Isbell and natural topologies. In general, the function spaces differ between the categories, but those of (vi) coincide with those in any of the larger categories (ii)–(v). Finally, the topologies of the spaces in the categories (i)–(iv) are analysed in terms of Lawson duality

A convenient category of locally stratified spaces

In this thesis we define the notion of a locally stratified space. Locally stratified spaces are particular kinds of streams and d-spaces which are locally modelled on stratified spaces. We construct a locally presentable and cartesian closed category of locally stratified spaces that admits an adjunction with the category of simplicial sets. Moreover, we show that the full subcategory spanned by locally stratified spaces whose associated simplicial set is an ∞-category has the structure of a category with fibrant objects. We define the fundamental category of a locally stratified space and show that the canonical functor θ_A from the fundamental category of a simplicial set A to the fundamental category of its realisation is essentially surjective. We show that the functor θ_A sends split monomorphisms to isomorphisms, in particular we show that θ_A is not necessarily an equivalence of categories. On the other hand, we show that the fundamental category of the realisation of the simplicial circle is equivalent to the monoid of the natural numbers. To conclude, we define left covers of locally stratified spaces and we show that, under suitable assumptions, the category of representations of the fundamental category of a simplicial set is equivalent to the category of left covers over its realisation

Epireflective subcategories of TOP, T 2 UNIF, UNIF, closed under epimorphic images, or being algebraic

The epireflective subcategories of Top, that are closed under epimorphic (or bimorphic) images, are { X∣ | X| ≤ 1 } , { X∣ X is indiscrete} and Top. The epireflective subcategories of T2Unif, closed under epimorphic images, are: { X∣ | X| ≤ 1 } , { X∣ X is compact T2} , { X∣ covering character of X is ≤ λ0} (where λ0 is an infinite cardinal), and T2Unif. The epireflective subcategories of Unif, closed under epimorphic (or bimorphic) images, are: { X∣ | X| ≤ 1 } , { X∣ X is indiscrete} , { X∣ covering character of X is ≤ λ0} (where λ0 is an infinite cardinal), and Unif. The epireflective subcategories of Top, that are algebraic categories, are { X∣ | X| ≤ 1 } , and { X∣ X is indiscrete}. The subcategories of Unif, closed under products and closed subspaces and being varietal, are { X∣ | X| ≤ 1 } , { X∣ X is indiscrete} , { X∣ X is compact T2}. The subcategories of Unif, closed under products and closed subspaces and being algebraic, are { X∣ X is indiscrete} , and all epireflective subcategories of { X∣ X is compact T2}. Also we give a sharpened form of a theorem of Kannan-Soundararajan about classes of T3 spaces, closed for products, closed subspaces and surjective images. © 2016, Akadémiai Kiadó, Budapest, Hungary