We provide a complete classification, up to order-isomorphism, of all possible Wadge hierarchies on zero-dimensional Polish spaces using (essentially) countable ordinals as complete invariants. We also observe that although our assignment of invariants is very simple and there are only $ \aleph _1 $-many equivalence classes, the above classification problem is quite complex from the descriptive set-theoretic point of view: in particular, there is no Borel procedure to determine whether two zero-dimensional Polish spaces have isomorphic Wadge hierarchies. All results are based on a complete and explicit description of the Wadge hierarchy on an arbitrary zero-dimensional Polish space, depending on its topological properties.

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

In this note, we show that in a complete $\operatorname {\mathrm {CAT}}(0)$ space pointwise convergence of proximal mappings under a certain normalization condition implies Mosco convergence.

We study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely $n$ types of countable dense sets: such a space contains a subset $S$ of size at most $n-1$ such that $S$ is invariant under all homeomorphisms of $X$ and $X\,\backslash \,S$ is countable dense homogeneous. We prove that every Borel space having fewer than $\mathfrak{c}$ types of countable dense sets is Polish. The natural question of whether every Polish space has either countably many or $\mathfrak{c}$ many types of countable dense sets is shown to be closely related to Topological Vaught's Conjecture.

In this paper we study the Stone-Čech bicompactification () of the bispace (X, P, Q). We show that the ring of all continuous real-valued functions on () may be identified with the uniform closure of a suitable subring of C(). Using this result, we give a characterization of the Wallman-Sanin compactifications of the pairwise Tychonoff bitopological spaces.

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?

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.

A function f:X → Y is almost continuous if for every x ∊ X and for each open set V ⊂ Y containing f(x), Cl(f-l(V)) is a neighborhood of x. Various conditions are given that guarantee that an almost continuous function is continuous. The main theorem states that if f:X → Y is almost continuous with a closed graph (closed in X × Y) and X and Y are complete metric spaces, then f is continuous.