Related papers: Compact spaces generated by retractions

Let R denote the smallest class of compact spaces containing all metric compacta and closed under limits of continuous inverse sequences of retractions. Class R is striclty larger than the class of Valdivia compact spaces. We show that…

We prove a preservation theorem for the class of Valdivia compact spaces, which involves inverse sequences of ``simple'' retractions. Consequently, a compact space of weight $\loe\aleph_1$ is Valdivia compact iff it is the limit of an…

We investigate the class of continuous images of non-commutative Valdivia compact spaces, in particular its subclass of weakly non-commutative Cor- son countably compact spaces. A key tool is the study of non-commutative Corson countably…

It is know that the Valdivia compact spaces can be characterized by a special family of retractions called $r$-skeleton (see \cite{kubis1}). Also we know that there are compact spaces with $r$-skeletons which are not Valdivia. In this…

We investigate the class of compact spaces which are embeddable into a power of the real line $R^\kappa$ in such a way that c_0(\kappa) is dense in the image. We show that this is a proper subclass of the class of Valdivia, even when…

We give a new characterization of Valdivia compact spaces: A compact space is Valdivia if and only if it has a dense commutatively monotonically retractable subspace. This result solves Problem 5.12 from \cite{sal-rey}. Besides, we…

An $r$-skeleton on a compact space is a family of continuous retractions having certain rich properties. The $r$-skeletons have been used to characterized the Valdivia compact spaces and the Corson compact spaces. Here, we characterized a…

We prove some generalizations of results concerning Valdivia compact spaces (equivalently spaces with a commutative retractional skeleton) to the spaces with a retractional skeleton (not necessarily commutative). Namely, we show that the…

It follows from a theorem of Rosenthal that a compact space is $ccc$ if and only if every Eberlein continuous image is metrizable. Motivated by this result, for a class of compact spaces $\mathcal{C}$ we define its orthogonal…

We study linearly ordered spaces which are Valdivia compact in their order topology. We find an internal characterization of these spaces and we present a counter-example disproving a conjecture posed earlier by the first author. The…

For any cardinal number $\kappa$ and an index set $\Gamma$, $\Sigma_\kappa$-product of real lines consists of elements of ${\mathbb R}^\Gamma$ having $<\kappa$ nonzero coordinates. A compact space $K$ is $\kappa$-Corson compact if it can be…

We construct a compact linearly ordered space $K$ of weight aleph one, such that the space $C(K)$ is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, $K$ is a continuous image of a…

We prove that every fragmentable linearly ordered compact space is almost totally disconnected. This combined with a result of Arvanitakis yields that every linearly ordered quasi Radon-Nikodym compact space is Radon-Nikodym, providing a…

We deeply study retractions associated to suitable models in compact spaces admitting a retractional skeleton and find several interesting consequences. Most importantly, we provide a new characterization of Valdivia compacta using the…

A compact space $X$ is said to be minimal if there exists a map $f:X\to X$ such that the forward orbit of any point is dense in $X$. We consider rigid minimal spaces, motivated by recent results of Downarowicz, Snoha, and Tywoniuk [J. Dyn.…

We extend the celebrated rigidity of the sharp first spectral gap under $Ric\ge0$ to compact infinitesimally Hilbertian spaces with non-negative (weak, also called synthetic) Ricci curvature and bounded (synthetic) dimension i.e. to…

In this article, we mainly study certain families of continuous retractions ($r$-skeletons) having certain rich properties. By using monotonically retractable spaces we solve a question posed by R. Z. Buzyakova in \cite{buz} concerning the…

Given a compact space in a fixed universe of set theory, one can naturally define its interpretation in any ZFC extension of the universe. We investigate the stability of some classes of compact spaces with respect to extensions of this…

We describe a family of hyperplane arrangements depending on a positive integer parameter $r$, which we refer to as the $r$-braid arrangements, and which can be viewed as a generalization of the classical braid arrangement. The wonderful…

We prove that every point-finite family of nonempty functionally open sets in a topological space $X$ has the cardinality at most an infinite cardinal $\kappa$ if and only if $w(X)\leq\kappa$ for every Valdivia compact space $Y\subseteq…