We prove that the opposite of the category of coalgebras for the Vietoris endofunctor on the category of compact Hausdorff spaces is monadic over $\mathsf {Set}$. We deliver an analogous result for the upper, lower, and convex Vietoris endofunctors acting on the category of stably compact spaces. We provide axiomatizations of the associated (infinitary) varieties. This can be seen as a version of Jónsson–Tarski duality for modal algebras beyond the zero-dimensional setting.

An old question of Arhangel’skii asks if the Menger property of a Tychonoff space X is preserved by homeomorphisms of the space $C_p(X)$ of continuous real-valued functions on X endowed with the pointwise topology. We provide affirmative answer in the case of linear homeomorphisms. To this end, we develop a method of studying invariants of linear homeomorphisms of function spaces $C_p(X)$ by looking at the way X is positioned in its (Čech–Stone) compactification.

Erdős space $\mathfrak {E}$ and complete Erdős space $\mathfrak {E}_{c}$ have been previously shown to have topological characterizations. In this paper, we provide a topological characterization of the topological space $\mathbb {Q}\times \mathfrak {E}_{c}$ , where $\mathbb {Q}$ is the space of rational numbers. As a corollary, we show that the Vietoris hyperspace of finite sets $\mathcal {F}(\mathfrak {E}_{c})$ is homeomorphic to $\mathbb {Q}\times \mathfrak {E}_{c}$ . We also characterize the factors of $\mathbb {Q}\times \mathfrak {E}_{c}$ . An interesting open question that is left open is whether $\sigma \mathfrak {E}_{c}^{\omega }$ , the $\sigma $ -product of countably many copies of $\mathfrak {E}_{c}$ , is homeomorphic to $\mathbb {Q}\times \mathfrak {E}_{c}$ .

Let G be a locally compact group and let ${\mathcal {SUB}(G)}$ be the hyperspace of closed subgroups of G endowed with the Chabauty topology. The main purpose of this paper is to characterise the connectedness of the Chabauty space ${\mathcal {SUB}(G)}$ . More precisely, we show that if G is a connected pronilpotent group, then ${\mathcal {SUB}(G)}$ is connected if and only if G contains a closed subgroup topologically isomorphic to ${{\mathbb R}}$ .

Let $(X,T)$ be a topological dynamical system consisting of a compact metric space X and a continuous surjective map $T : X \to X$ . By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does the induced system $({\mathcal {M}}(X),\widetilde {T})$ on the space of Borel probability measures endowed with the weak* topology. This result can be seen as a version for the notion of uniformly positive entropy of the corresponding result for topological entropy due to Glasner and Weiss.

We establish that the existence of a winning strategy in certain topological games, closely related to a strong game of Choquet, played in a topological space $X$ and its hyperspace $K(X)$ of all nonempty compact subsets of $X$ equipped with the Vietoris topology, is equivalent for one of the players. For a separable metrizable space $X$, we identify a game-theoretic condition equivalent to $K(X)$ being hereditarily Baire. It implies quite easily a recent result of Gartside, Medini and Zdomskyy that characterizes hereditary Baire property of hyperspaces $K(X)$ over separable metrizable spaces $X$ via the Menger property of the remainder of a compactification of $X$. Subsequently, we use topological games to study hereditary Baire property in spaces of probability measures and in hyperspaces over filters on natural numbers. To this end, we introduce a notion of strong $P$-filter ${\mathcal{F}}$ and prove that it is equivalent to $K({\mathcal{F}})$ being hereditarily Baire. We also show that if $X$ is separable metrizable and $K(X)$ is hereditarily Baire, then the space $P_{r}(X)$ of Borel probability Radon measures on $X$ is hereditarily Baire too. It follows that there exists (in ZFC) a separable metrizable space $X$, which is not completely metrizable with $P_{r}(X)$ hereditarily Baire. As far as we know, this is the first example of this kind.

In this paper we provide a unified approach, based on methods of descriptive set theory, for proving some classical selection theorems. Among them is the zero-dimensional Michael selection theorem, the Kuratowski–Ryll-Nardzewski selection theorem, as well as a known selection theorem for hyperspaces.

In this note, we study the recurrence and topologically multiple recurrence of a sequence of operators on Banach spaces. In particular, we give a sufficient and necessary condition for a cosine operator function, induced by a sequence of operators on the Lebesgue space of a locally compact group, to be topologically multiply recurrent.

An example is given of a map $f$ defined between arcwise connected continua such that $C(f)$ is light and ${{2}^{f}}$ is not light, giving a negative answer to a question of Charatonik and Charatonik. Furthermore, given a positive integer $n$, we study when the lightness of the induced map ${{2}^{f}}$ or ${{C}_{n}}(f)$ implies that $f$ is a homeomorphism. Finally, we show a result in relation with the lightness of $C(C(f))$.

In this paper we use the Hausdorff metric to prove that two compact metric spaces are homeomorphic if and only if their canonical complements are uniformly homeomorphic. We take one of the two steps needed to prove that the difference between the homotopical and topological classifications of compact connected ANRs depends only on the difference between continuity and uniform continuity of homeomorphisms in their canonical complements, which are totally bounded metric spaces. The more important step was provided by the Chapman complement and the Curtis–Schori–West theorems. We also improve the multivalued description of shape theory given by J. M. R. Sanjurjo but only in the class of locally connected compacta.

The notion of expansive homeomorphism is important in topological dynamics and continuum theory. In this paper, a new kind of homeomorphism will be introduced and studied, namely the continuum-wise expansive homeomorphism. The class of continuum-wise expansive homeomorphisms is much larger than the one of expansive homeomorphisms. In fact, the class of continuum-wise expansive homeomorphisms contains many important homeomorphisms which often appear in "chaotic" topological dynamics and continuum theory, but which are not expansive homeomorphisms. For example, the shift maps of Knaster's indecomposable chainable continua are continuum-wise expansive homeomorphisms, but they are not expansive homeomorphisms. Also, there is a continuum-wise expansive homeomorphism on the pseudoarc. We study several properties of continuum-wise expansive homeomorphisms. Many theorems concerning expansive homeomorphisms will be generalized to the case of continuum-wise expansive homeomorphisms.

A new topology for the closed subsets of a topological space X which are the closure of their interiors is defined and investigated. Some applications to convergence of regular measures are also given.

Let X be a complete metric space. Viewing continuous real functions on X as closed subsets of X × R, equipped with Hausdorff distance, we show that C(X, R) is completely metrizable provided X is complete and sigma compact. Following the Bulgarian school of constructive approximation theory, a bounded discontinuous function may be identified with its completed graph, the set of points between the upper and lower envelopes of the function. We show that the space of completed graphs, too, is completely metrizable, provided X is locally connected as well as sigma compact and complete. In the process, when X is a Polish space, we provide a simple answer to the following foundational question: which subsets of X × R arise as completed graphs?

In proving the contractibility of certain hyperspaces J. L. Kelley identified and defined a certain uniformnessproperty which he called Property 3.2. It is known that the classes of locally connected continua, homogeneous continua and hereditarily indecomposable continua have Property 3.2. In this paper we prove that two examples of indecomposable continua developed respectively by the authors have Property 3.2. One is the example of a nonchainable atriodic tree-like continuum with positive span which was defined by the first author, and the other is a nonchainable, noncircle-like continuum which has the cone=hyperspace property which was defined by the second author. Each of the examples is an inverse limit of an inverse system having a single bonding map.

Let 〈X, dx〉 and 〈Y, dY〉 be metric spaces and let hp denote Hausdorff distance in X x Y induced by the metric p on X x Y given by p[(x1, y1), (x2, y2)] = max ﹛dx(x1, x2),dY(y1, y2)﹜- Using the fact that hp when restricted to the uniformly continuous functions from X to Y induces the topology of uniform convergence, we exhibit a natural compactness criterion for C(X, Y) when X is compact and Y is complete.

Let C(X, Y) denote the set of continuous functions from a metric space X to a metric space Y. Viewing elements of C(X, Y) as closed subsets of X × Y, we say {fn} converges topologically to f if Li fn = Lsfn = f. If X is connected, then topological convergence in C(X,R) does not imply pointwise convergence, but if X is locally connected and Y is locally compact, then topological convergence in C(X, Y) is equivalent to uniform convergence on compact subsets of X. Pathological aspects of topological convergence for seemingly nice spaces are also presented, along with a positive Baire category result.

Let X and Y be metric spaces. This paper considers the relationship between uniform convergence in C(X, Y) and topological convergence of functions in C(X, Y), viewed as subsets of X×Y. In general, uniform convergence in C(X, Y) implies Hausdorff metric convergence which, in turn, implies topological convergence, but if X and Y are compact, then all three notions are equivalent. If C([0, 1], Y) is nontrivial arid topological convergence in C(X, Y) implies uniform converger ce then X is compact. Theorem: Let X be compact and Y be loyally compact but noncompact. Then topological convergence in C(X, Y) implies uniform convergence if and only if X has finitely many components. We also sharpen a related result of Naimpally.

If X is a topological space then exp X denotes the space of non-empty closed subsets of X with the Vietoris topology and λX denotes the superextension of X Using Martin's axiom together with the negation of the continuum hypothesis the following is proved: If every discrete subset of exp X is countable the X is compact and metrizable. As a corollary, if λX contains no uncountable discrete subsets then X is compact and metrizable. A similar argument establishes the metrizability of any compact space X whose square X × X contains no uncountable discrete subsets.