183,484 research outputs found
On dimensions modulo a compact metric ANR and modulo a simplicial complex
V. V. Fedorchuk has recently introduced dimension functions K-dim \leq K-Ind
and L-dim \leq L-Ind, where K is a simplicial complex and L is a compact metric
ANR. For each complex K with a non-contractible join |K| * |K| (we write |K|
for the geometric realisation of K), he has constructed first countable,
separable compact spaces with K-dim < K-Ind.
In a recent paper we have combined an old construction by P. Vop\v{e}nka with
a new construction by V. A. Chatyrko, and have assigned a certain compact space
Z (X, Y) to any pair of non-empty compact spaces X, Y. In this paper we
investigate the behaviour of the four dimensions under the operation Z (X, Y).
This enables us to construct more examples of compact Fr\'echet spaces which
have prescribed values K-dim < K-Ind, L-dim < L-Ind, or K-Ind < |K|-Ind, and
(connected) components of which are metrisable.Comment: 24 pages, 1 figur
Complementation and decompositions in some weakly Lindelöf Banach spaces
AbstractLet Î denote an uncountable set. We consider the questions if a Banach space X of the form C(K) of a given class (1) has a complemented copy of c0(Î) or (2) for every c0(Î)âX has a complemented c0(E) for an uncountable EâÎ or (3) has a decomposition X=AâB where both A and B are nonseparable. The results concern a superclass of the class of nonmetrizable Eberlein compacts, namely Ks such that C(K) is Lindelöf in the weak topology and we restrict our attention to Ks scattered of countable height. We show that the answers to all these questions for these C(K)s depend on additional combinatorial axioms which are independent of ZFC±CH. If we assume the P-ideal dichotomy, for every c0(Î)âC(K) there is a complemented c0(E) for an uncountable EâÎ, which yields the positive answer to the remaining questions. If we assume âŁ, then we construct a nonseparable weakly Lindelöf C(K) for K of height Ï+1 where every operator is of the form cI+S for câR and S with separable range and conclude from this that there are no decompositions as above which yields the negative answer to all the above questions. Since, in the case of a scattered compact K, the weak topology on C(K) and the pointwise convergence topology coincide on bounded sets, and so the Lindelöf properties of these two topologies are equivalent, many results concern also the space Cp(K)
Chain conditions and weak topologies
AbstractWe study conditions on Banach spaces close to separability. We say that a topological space is pcc if every point-finite family of open subsets of the space is countable. For a Banach space E, we say that E is weakly pcc if E, equipped with the weak topology, is pcc, and we also consider a weaker property: we say that E is half-pcc if every point-finite family consisting of half-spaces of E is countable. We show that E is half-pcc if, and only if, every bounded linear map Eâc0(Ï1) has separable range. We exhibit a variety of mild conditions which imply separability of a half-pcc Banach space. For a Banach space C(K), we also consider the pcc-property of the topology of pointwise convergence, and we note that the space Cp(K) may be pcc even when C(K) fails to be weakly pcc. We note that this does not happen when K is scattered, and we provide the following example:âThere exists a non-metrizable scattered compact Hausdorff space K with C(K) weakly pcc
Joins for (Augmented) Simplicial Sets
We introduce a notion of join for (augmented) simplicial sets generalising
the classical join of geometric simplicial complexes. The definition comes
naturally from the ordinal sum on the base simplicial category .Comment: 8 page
Business Regulation in International Comparison â Aggregating World Bank âDoing Businessâ Data
Unternehmensregulierung, Management, Rangstatistik, Vergleich, Regulated firm, Comparison
- âŠ