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We investigate the solvability in continuous functions of the Abel equation φ(Fx) - φ(x) = 1 where F is a given continuous mapping of a topological space X. This property depends on the dynamics generated by F. The solvability of all linear equations P(x)ψ(Fx) + Q(x)ψ(x) = γ(x) follows from solvability of the Abel equation in case F is a homeomorphism. If F is noninvertible but X is locally compact then such a total solvability is determined by the same property of the cohomological equation φ(Fx)...
Answering a question of Halbeisen we prove (by two different methods) that the algebraic dimension of each infinite-dimensional complete linear metric space X equals the size of X. A topological method gives a bit more: the algebraic dimension of a linear metric space X equals |X| provided the hyperspace K(X) of compact subsets of X is a Baire space. Studying the interplay between Baire properties of a linear metric space X and its hyperspace, we construct a hereditarily Baire linear metric space...
We characterize metric spaces whose hyperspaces of non-empty closed, bounded, compact or finite subsets, endowed with the Attouch-Wets topology, are absolute (neighborhood) retracts.
We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel’skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in , and a version of an (ω,1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.
We show that MA implies that normal locally compact metacompact spaces are paracompact, and that MA() implies normal locally compact metalindelöf spaces are paracompact. The latter result answers a question of S. Watson. The first result implies that there is a model of set theory in which all normal locally compact metacompact spaces are paracompact, yet there is a normal locally compact metalindelöf space which is not paracompact.
Let , and be the spaces of all non-empty closed convex sets in a normed linear space X admitting the Hausdorff metric topology, the Attouch-Wets topology and the Wijsman topology, respectively. We show that every component of and the space are AR. In case X is separable, is locally path-connected.
Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every -space, hence any k-space, is Ascoli.
Let X be a metrizable space. We prove that the space is Ascoli iff is a -space iff X is locally compact. Moreover, endowed with the weak topology is Ascoli iff X is countable and discrete.
Using some basic concepts from probability theory and...
We show that the automorphism group Aut([0,1],λ) of the Lebesgue measure has no non-trivial subgroups of index .
It is established that a remainder of a non-locally compact topological group has the Baire property if and only if the space is not Čech-complete. We also show that if is a non-locally compact topological group of countable tightness, then either is submetrizable, or is the Čech-Stone remainder of an arbitrary remainder of . It follows that if and are non-submetrizable topological groups of countable tightness such that some remainders of and are homeomorphic, then the spaces...
We prove a commutative Gelfand-Naimark type theorem, by showing that the set of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space X (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to C₀(Y) for some unique (up to homeomorphism) locally compact Hausdorff space Y. The space Y, which we explicitly construct as a subspace of the Stone-Čech compactification of X, is countably compact, and if X is non-separable,...
By a dynamical system we mean the action of the semigroup on a metrizable topological space induced by a continuous selfmap . Let denote the set of all compatible metrics on the space . Our main objective is to show that a selfmap of a compact space is a Banach contraction relative to some if and only if there exists some which, regarded as a -cocycle of the system , is a coboundary.
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