The Sorgenfrey line has no connected compactification
The space of Whitney levels is homeomorphic to
The span and the width of continua
The structure of atoms (hereditarily indecomposable continua)
Let X be an atom (= hereditarily indecomposable continuum). Define a metric ϱ on X by letting where is the (unique) minimal subcontinuum of X which contains x and y and W is a Whitney map on the set of subcontinua of X with W(X) = 1. We prove that ϱ is an ultrametric and the topology of (X,ϱ) is stronger than the original topology of X. The ϱ-closed balls C(x,r) = y ∈ X:ϱ ( x,y) ≤ r coincide with the subcontinua of X. (C(x,r) is the unique subcontinuum of X which contains x and has Whitney value...
The superextension of the closed unit interval is homeomorphic to the Hilbert cube
The Suslinian number and other cardinal invariants of continua
By the Suslinian number Sln(X) of a continuum X we understand the smallest cardinal number κ such that X contains no disjoint family ℂ of non-degenerate subcontinua of size |ℂ| > κ. For a compact space X, Sln(X) is the smallest Suslinian number of a continuum which contains a homeomorphic copy of X. Our principal result asserts that each compact space X has weight ≤ Sln(X)⁺ and is the limit of an inverse well-ordered spectrum of length ≤ Sln(X)⁺, consisting of compacta with weight ≤ Sln(X) and...
The theorem of Miss Mullikin-Mazurkiewicz-van East for unicoherent Peano spaces
The topological fixed point property - an elementary continuum-theoretic approach
A set contained in a topological space has the topological fixed point property if every continuous mapping of the set into itself leaves some point fixed. In 1969, R. H. Bing published his article The Elusive Fixed Point Property, posing twelve intriguing and difficult problems, which exerted a great influence on the study of the fixed point property. We now present a survey article intended for a broad audience that reports on this area of fixed point theory. The exposition is also intended to...
The Whitehead Theorem in the theory of shapes
Three questions of Borsuk concerning movability and fundamental retraction
Topological spaces with a linear basis
Toroidal decompositions of and a family of 3-dimensional ANR’s (AR’s)
Tree-likeness of hereditarily equivalent continua
Two-to-one maps on solenoids and Knaster continua
It is shown that 2-to-1 maps cannot be defined on certain solenoids, in particular on the dyadic solenoid, and on Knaster continua.