Mathematics

It is given a simplified and self-contained proof of the classical Michael's finite-dimensional selection theorem. The proof is based on approximate selections constructed stepwise over skeletons of nerves of covers. The method is also applied to simplify the proof of the Schepin--Brodsky's generalisation of this theorem.

For any compact Hausdorff space K we construct a canonical finitary coarse structure E X,K on the set X of isolated points of K . This construction has two properties: ∙ If a finitary coarse space (X,E) is metrizable, then its coarse structure E coincides with the coarse structure E X, X ¯ generated by the Higson compactification X ¯ of X ; ∙ A compact Hausdorff space K coincides with the Higson compactification of the coarse space (X, E X,K ) if the set X is dense in K and the space K is Frechet-Urysohn. This implies that a compact Hausdorff space K is homeomorphic to the Higson corona of some finitary coarse space if one of the following conditions holds: (i) K is perfectly normal; (ii) K has weight w(K)≤ ω 1 and character χ(K)<p . Under CH every (zero-dimensional) compact Hausdorff space of weight ≤ ω 1 is homeomorphic to the Higson (resp. binary) corona of some cellular finitary coarse space.

In 1972, Dana Scott proved a fundamental result on the connection between order and topology which says that injective T 0 spaces are precisely continuous lattices endowed with Scott topology. This paper investigates whether this is true in the enriched context, where the enrichment is a quantale obtained by equipping the interval [0,1] with a continuous t-norm. It is shown that for each continuous t-norm, the specialization [0,1] -order of an injective and separated [0,1] -approach space X is a continuous [0,1] -lattice and the [0,1] -approach structure of X coincides with the Scott [0,1] -approach structure of the specialization [0,1] -order; but, unlike in the classical situation, the converse fails in general.

A space X is strongly Y -selective (resp., Y -selective) if every lower semicontinuous mapping from Y to the nonempty subsets (resp., nonempty closed subsets) of X has a continuous selection. We also call X (strongly) C -selective if it is (strongly) Y -selective for any countable space Y , and (strongly) L -selective if it is (strongly) ( ω+1 )-selective. E. Michael showed that every first countable space is strongly C -selective. We extend this by showing that every W -space in the sense of the second author is strongly C -selective. We also show that every GO-space is C -selective, and that every L -selective space has Arhangel'skii's property α 1 . We obtain an example under p=c of a strongly L -selective space that is not C -selective, and we show that it is consistent with and independent of ZFC that a space is strongly L -selective iff it is L -selective and Fréchet. Finally, we answer a question of the third author and Junnila by showing that the ordinal space ω 1 +1 is not self-selective.

The contour of a family of filters along a filter is a set-theoretic lower limit. Topologicity and regularity of convergences can be characterized with the aid of the contour operation. Contour inversion is studied, in particular, for iterated contours of sequential cascades. A related problem of continuous extension of maps between maximal elements of sequential cascades to full subcascades is solved in full generality.

Let (X,τ) be a Hausdorff space and n∈ω . We prove that if X admits a continuous selection over F n (X) (nonempty subsets of X of cardinality at most n ), then for every n≤m≤2n such that m is not a prime number, X admits a continuous selection over [X ] m (subsets of X of cardinality m ). As a consequence of this, a space X admits a continuous selection for every natural number if and only if the same is true for every prime number. For Hausdorff spaces (X,τ) which admit continuous selections over [X ] 2 , we characterize the existence of continuous selections over [X ] n for n≥2 , in terms of a covering-type property.

For any composant E⊂ H ∗ and corresponding near-coherence class E⊂ ω ∗ we prove the following are equivalent : (1) E properly contains a dense semicontinuum. (2) Each countable subset of E is contained in a dense proper semicontinuum of E . (3) Each countable subset of E is disjoint from some dense proper semicontinuum of E . (4) E has a minimal element in the finite-to-one monotone order of ultrafilters. (5) E has a Q -point. A consequence is that NCF is equivalent to H ∗ containing no proper dense semicontinuum and no non-block points. This gives an axiom-contingent answer to a question of the author. Thus every known continuum has either a proper dense semicontinuum at every point or at no points. We examine the structure of indecomposable continua for which this fails, and deduce they contain a maximum semicontinuum with dense interior.

In this paper, we introduce a generalization of rectangular b− metric spaces, by changing the rectangular inequality as follows ρ(x,y)≤θ(x,y,u,v)[ρ(x,u)+ρ(u,v)+ρ(v,y)], % for all distinct x,y,u,v∈X. We prove some fixed-point theorems and we use our results to present a nice application in last section of this paper. Moreover, in the conclusion we present some new open questions.

In this paper, we first define the concept of convexity in G-metric spaces. We then use Mann iterative process in this newly defined convex G-metric space to prove some convergence results for some classes of mappings. In this way, we can extend several existence results to those approximating fixed points. Our results are just new in the setting.

In this paper, we introduce coproducts of proximity spaces. After exploring several of their basic properties, we show that given a collection of proximity spaces, the coproduct of their Smirnov compactifications proximally and densely embeds in the Smirnov compactification of the coproduct of the original proximity spaces. We also show that the dense proximity embedding is a proximity isomorphism if and only if the index set is finite. After constructing a number of examples of coproducts and their Smirnov compactifications, we explore several properties of the Smirnov compactification of the coproduct, including its metrizability, connectedness of the boundary, dimension, and its relation to the Stone-Cech compactification. In particular, we show that the Smirnov compactification of the infinite coproduct is never metrizable and that its boundary is highly disconnected. We also show that the proximity dimension of the Smirnov compactification of the coproduct equals the supremum of the covering dimensions of the individual Smirnov compactifications and that the Smirnov compactification of the coproduct is homeomorphic to the Stone-Cech compactification if and only if each individual proximity space is equipped with the Stone-Cech proximity. We finish with an example of a coproduct with the covering dimension 0 but the proximity dimension ∞.