Monotone retracts of an arcwise connected continuum
More on ordinals in topological groups
Let be an uncountable regular cardinal and a topological group. We prove the following statements: (1) If is homeomorphic to a closed subspace of , is Abelian, and the order of every non-neutral element of is greater than then embeds in as a closed subspace. (2) If is Abelian, algebraically generated by , and the order of every element does not exceed then is not embeddable in . (3) There exists an Abelian topological group such that is homeomorphic to a closed subspace...
Movability and shape-connectivity
Multicoherent spaces
Mutational retracts and extensions of mutations
n-Arc connected spaces
A space is n-arc connected (n-ac) if any family of no more than n-points are contained in an arc. For graphs the following are equivalent: (i) 7-ac, (ii) n-ac for all n, (iii) continuous injective image of a closed subinterval of the real line, and (iv) one of a finite family of graphs. General continua that are ℵ₀-ac are characterized. The complexity of characterizing n-ac graphs for n = 2,3,4,5 is determined to be strictly higher than that of the stated characterization of 7-ac graphs.
Natural continuity space structures on dual Heyting algebras
Necessary and sufficient conditions for unique determination of plane domains.
Neighborhoods of compacta in euclidean space
No arc-connected treelike continuum is the 2-to-1 image of a continuum
In 1940, O. G. Harrold showed that no arc can be the exactly 2-to-1 continuous image of a metric continuum, and in 1947 W. H. Gottschalk showed that no dendrite is a 2-to-1 image. In 2003 we show that no arc-connected treelike continuum is the 2-to-1 image of a continuum.
Non-additivity of the fixed point property for tree-like continua
We investigate the fixed point property for tree-like continua that are unions of tree-like continua. We obtain a positive result if finitely many tree-like continua with the fixed point property have dendrites for pairwise intersections. Using Bellamy's seminal example, we define (i) a countable wedge X̂ of tree-like continua, each having the fpp, and X̂ admitting a fixed-point-free homeomorphism, and (ii) two tree-like continua H and K such that H, K, and H∩ K have the fixed point property, but...
Non-alternating mappings
Nonexistence of local expansions on certain continua
Nonhyperbolic one-dimensional invariant sets with a countably infinite collection of inhomogeneities
We examine the structure of countable closed invariant sets under a dynamical system on a compact metric space. We are motivated by a desire to understand the possible structures of inhomogeneities in one-dimensional nonhyperbolic sets (inverse limits of finite graphs), particularly when those inhomogeneities form a countable set. Using tools from descriptive set theory we prove a surprising restriction on the topological structure of these invariant sets if the map satisfies a weak repelling or...
Nonisomorphic thin-tall superatomic Boolean algebras
Non-planar embeddings of planar sets in
Non-separating subcontinua of planar continua
We revisit an old question of Knaster by demonstrating that each non-degenerate plane hereditarily unicoherent continuum X contains a proper, non-degenerate subcontinuum which does not separate X.
Nonstandard analysis and the theory of shape
Normal P-spaces and the -topology
Normality and paracompactness of Pixley-Roy hyperspaces