Weak Hausdorff Research Articles

Goodwillie’s cosimplicial model for the space of long knots and its applications

We work out the details of a correspondence observed by Goodwillie between cosimplicial spaces and good functors from a category of open subsets of the interval to the category of compactly generated weak Hausdorff spaces. Using this, we compute the first page of the integral Bousfield–Kan homotopy spectral sequence of the tower of fibrations, given by the Taylor tower of the embedding functor associated to the space of long knots. Based on the methods in Conant (Am J Math 130(2):341–357. https://doi.org/10.1353/ajm.2008.0020, 2008), we give a combinatorial interpretation of the differentials d^1 mapping into the diagonal terms, by introducing the notion of (i, n)-marked unitrivalent graphs.

Generic theorems in the theory of cardinal invariants of topological spaces

<p>The main aim of this paper is to present a technical result, which provides an algorithm to prove several cardinal inequalities and relative versions of cardinal inequalities related. Moreover, we use this result and the weak Hausdorff number, H<sup>∗</sup>, introduced by Bonanzinga in [Houston J. Math. 39 (3) (2013), 1013–1030], to generalize some upper bounds on the cardinality of topological spaces.</p>

Krein-Smulian-Type Theorems

AbstractLet (E, ℱ) be a weakly compactly generated Frechet space and let ℱ0 be another weaker Hausdorff locally convex topology on E. Let X be an ℱ-bounded compact subset of (E, ℱ0). The ℱ0-closed convex hull of X in E is then ℱ0-compact. We also give a new proof, without using Riemann–Lebesgue-integrable (Birkoff-integrable) functions, with the result that if (E, ∥ · ∥) is any Banach space and ℱ0 is fragmented by ∥ · ∥, then the same result holds. Furthermore, the closure of the convex hull of X in ℱ0-topology and in the original topology of E is the same.

Weak Hausdorff separation axiom in I-fuzzy topological spaces and its application

In this paper, the weak Hausdorff separation axiom in the sense of Fang and Ren [A set of new separations axioms in L-fuzzy topological spaces, Fuzzy Sets and Systems 96 (1998) 359–376] is generalized from L-topologies to I-fuzzy topologies. It is shown that this notion satisfies the hereditary, productive properties, and the relations between it and other separation axioms are obtained. Also the degree to which an I-fuzzy topological space is weak Hausdorff is studied in terms of the fuzzy nets and the corresponding level I-topologies. As an application, we prove that the degree to which an I-fuzzy topological group is weak Hausdorff is equal to the degree to which it is T 1.

Many homotopy categories are homotopy categories

We show that any category that is enriched, tensored, and cotensored over the category of compactly generated weak Hausdorff spaces, and that satisfies an additional hypothesis concerning the behavior of colimits of sequences of cofibrations, admits a Quillen closed model structure in which the weak equivalences are the homotopy equivalences. The fibrations are the Hurewicz fibrations and the cofibrations are a subclass of the Hurewicz cofibrations. This result applies to various categories of spaces, unbased or based, categories of prespectra and spectra in the sense of Lewis and May, the categories of L -spectra and S-modules of Elmendorf, Kriz, Mandell and May, and the equivariant analogues of all the afore-mentioned categories.

Weak Hausdorff Gaps and the 𝔭 ≤ t Problem

We find topological characterizations of the pseudointersection number 𝔭 and the tower number t of the real line and we show that 𝔭 < t iff there exists a compact separable T2 space X of π-weight < 𝔭 that can be covered by < t nowhere dense sets iff there exists a weak Hausdorff gap of size K < t, i. e., a pair ({A : i ≠ k}, {BJ : j ε K}) C [W]W X [U]W such that A = {Ai : i ε K} is a decreasing tower, B = {Bj : j ε K) is a family of pseudointersections of A, and there is no pseudointersection S of A meeting each member of B in an infinite set.

On spaces in which every bounded subset is Hausdorff

The class of spaces in the title (denoted by Haus(e-comp)) is introduced and it is compared with other classes of weak Hausdorff spaces. An explicit description of the Haus(e-comp)- epimorphisms is given by means of a variation of Arhangel'skii–Franklin's compactly determined closure. It is shown that Haus(e-comp) is not co-well-powered.

Open maps, colimits, and a convenient category of fibre spaces

We show that pulling back along an open map preserves all colimits in the category of weak Hausdorff k-spaces. We also show that the category of open maps over a weak Hausdorff k-space is a convenient category of fibre spaces.

Covering homotopy properties of maps between C.W. complexes or ANRs

Serre fibrations between C.W. complexes are fibrations in the category of compactly generated weak Hausdorff spaces, but not, by example, in the category of all spaces. Under suitable hypotheses the notions of Dold fibration in the two categories coincide. Examples are given of maps with ANR base, total space and fibres which are not Dold fibrations but whose pullbacks over maps out of locally finite-dimensional paracompact spaces are bundles.