A space X is cometrizable if X has a coarser metric topology such that each point of X has a neighborhood base of metric closed sets. Most examples in the literature of spaces obtained by modifying the topology of the plane or some other metric space are cometrizable. Assuming the Proper Forcing Axiom (PFA) we show that the following statements are equivalent for a cometrizable space X : (a) X ...

Let X and Y be separable Banach spaces. Suppose Y either has a shrinking basis or Y is isomorphic to C(2N) andA is a subset of weakly compact operators from X to Y which is analytic in the strong operator topology. We prove that there is a reflexive space with a basis Z such that every T ∈ A factors through Z. Likewise, we prove that if A ⊂ L(X,C(2N)) is a set of operators whose adjoints have s...

Given a map T : X → X on a set X we examine under what conditions there is a separable metrizable or an hereditarily Lindelöf or a Lindelöf topology on X with respect to which T is a continuous map. For separable metrizable and hereditarily Lindelöf, it turns out that there is a such a topology precisely when the cardinality ofX is no greater than the cardinality of the continuum. We go onto pr...

Given a map T : X → X on a set X we examine under what conditions there is a separable metrizable or an hereditarily Lindelöf or a Lindelöf topology on X with respect to which T is a continuous map. For separable metrizable and hereditarily Lindelöf, it turns out that there is such a topology precisely when the cardinality of X is no greater than c, the cardinality of the continuum. We go on to...

We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Ru...

LetM be a closed, orientable, irreducible, geometrizable 3-manifold. We prove that the profinite topology on the fundamental group of π1(M) is efficient with respect to the JSJ decomposition of M . We go on to prove that π1(M) is good, in the sense of Serre, if all the pieces of the JSJ decomposition are. We also prove that if M is a graph manifold then π1(M) is conjugacy separable. A group G i...

De Groot proved that every group is the autohomeomorphism group of some metrizable space. A space is totally disconnected if every connected subset of it contains at most one point. We prove that every separable metrizable totally disconnected topological group is topologically isomorphic to the autohomeomorphism group of some separable metrizable space, when given the compact-open topology. It...

For a regular space X, the hyperspace (CL(X),τF) (resp., (CL(X),τV)) is of all nonempty closed subsets X with Fell topology Vietoris topology). In this paper, we give characterization such that countable character subsets. We mainly prove has on each subset if and only compact metrizable, locally separable metrizable. Moreover, (K(X),τV) have compact-Gδ-property every

We prove that the fundamental group of any Seifert 3-manifold is conjugacy separable. That is, conjugates may be distinguished in finite quotients or, equivalently, conjugacy classes are closed in the pro-finite topology.