We prove that the classes of weakly $1$-dimensional and almost $0$-dimensional spaces are disjoint. The result has applications to hereditarily locally connected spaces, $\mathbb R$-trees, endpoints smooth fans.

We show that each of the classes of hereditarily locally connected, finitely Suslinian, and Suslinian continua is Π1-complete, while the class of regular continua is Π4-complete.

In this paper we characterize hereditarily-normal spaces in terms of the measure-theoretic properties of the lattice of closed sets. We then generalize from that lattice to other lattices. We apply the results to extremallydisconnected spaces.

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 ...

It is proved deenability in FO+IFP of a global linear ordering on vertices of strongly extensional (SE) nitely-branching graphs. In the case of nite SE graphs this also holds for FO+LFP. This gives capturing results for PTIME computability on the latter class of graphs by FO+LFP and FO+IFP, and also on the corresponding anti-founded universe HFA of hereditarily-nite sets by a language of a boun...

Results are obtained about the existence and behavior of hereditarily weakly confluent maps of continua onto the unit circle S1. A simple and useful necessary and sufficient condition is given for a map of a continuum, X, onto S1 to be hereditarily weakly confluent (HWC). It is shown that when X is arcwise connected, an HWC map of X onto S1 is monotone with arcwise connected fibers. A number of...