It is shown that for every integer n the (2n+1)th power of any locally path-connected metrizable space of the first Baire category is 𝓐₁[n]-universal, i.e., contains a closed topological copy of each at most n-dimensional metrizable σ-compact space. Also a one-dimensional σ-compact absolute retract X is found such that the power X^{n+1} is 𝓐₁[n]-universal for every n.
In this paper we consider rational subspaces of the plane. A rational space is a space which has a basis of open sets with countable boundaries. In the special case where the boundaries are finite, the space is called rim-finite.
Let be a continuum. Two maps are said to be pseudo-homotopic provided that there exist a continuum , points and a continuous function such that for each , and . In this paper we prove that if is the pseudo-arc, is one-to-one and is pseudo-homotopic to , then . This theorem generalizes previous results by W. Lewis and M. Sobolewski.
A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems...