A metrizable semitopological semilattice with non-closed partial order

Basic Intervals in the Partial Order of Metrizable Topologies

For a set X, let Σm(X) denote the set of metrizable topologies on X, partially ordered by inclusion. We investigate the nature of intervals in this partial order, with particular emphasis on basic intervals (in other words, intervals in which the topology changes at at most one point). We show that there are no non-trivial finite intervals in Σm(X) (indeed, every such interval contains a copy o...

The countable existentially closed pseudocomplemented semilattice

Weak Topologies for the Closed Subsets of a Metrizable Space

The purpose of this article is to propose a unified theory for topologies on the closed subsets of a metrizable space. It can be shown that all of the standard hyperspace topologies—including the Hausdorff metric topology, the Vietoris topology, the Attouch-Wets topology, the Fell topology, the locally finite topology, and the topology of Mosco convergence—arise as weak topologies generated by ...

Two More Perfectly Normal Non-metrizable Manifolds

We show that there is a perfectly normal non-metrizable manifold if there is a Luzin subset of the real line, and that there is a countably compact perfectly normal non-metrizable manifold in any model of set-theory obtained by adding Cohen reals to a model of ZFC +♦.

Homotopy in non metrizable ω-bounded surfaces

We investigate the problem of describing the homotopy classes [X,Y ] of continuous functions between ω-bounded non metrizable manifolds X,Y . We define a family of surfaces X built with the first octant C in L (L is the longline and R the longray), and show that [X,R] is in bijection with so called ‘adapted’ subsets of a partially ordered set. We also show that [M,R] can be computed for some su...