Let X and Y be two Polish spaces. Functions f,g: X → Y are called equivalent if there exists a bijection φ from X onto itself such that g∘φ = f. Using a theorem of J. Saint Raymond we characterize functions equivalent to Borel measurable ones. This characterization answers a question asked by M. Morayne and C. Ryll-Nardzewski.
Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1 functions...
A function mapping the topological space to the space is called a z-open function if for every cozeroset neighborhood of a zeroset in , the image is a neighborhood of in . We say has the z-separation property if whenever , are cozerosets and is a zeroset of such that , there is a zeroset of such that . A surjective function is z-open if and only if it maps cozerosets to cozerosets and has the z-separation property. We investigate z-open functions and other functions...
It is shown that if Ω = Q or Ω = ℓ 2, then there exists a functor of extension of maps between Z-sets in Ω to mappings of Ω into itself. This functor transforms homeomorphisms into homeomorphisms, thus giving a functorial setting to a well-known theorem of Anderson [Anderson R.D., On topological infinite deficiency, Michigan Math. J., 1967, 14, 365–383]. It also preserves convergence of sequences of mappings, both pointwise and uniform on compact sets, and supremum distances as well as uniform continuity,...
The aim of the paper is to prove that in the unbounded Urysohn universal space there is a functor of extension of -isometric maps (i.e. dilations) between central subsets of to -isometric maps acting on the whole space. Special properties of the functor are established. It is also shown that the multiplicative group acts continuously on by -isometries.
Following the paper [BDC1], further relations between the classical topologies on function spaces and new ones induced by hyperspace topologies on graphs of functions are introduced and further characterizations of boundedly spaces are given.
Some relationships between -sequence-covering maps and weak-open maps or sequence-covering -maps are discussed. These results are used to generalize a result from Lin S., Yan P., Sequence-covering maps of metric spaces, Topology Appl. 109 (2001), 301–314.
In this paper, we prove that a space is a -metrizable space if and only if is a weak-open, and -image of a semi-metric space, if and only if is a strong sequence-covering, quotient, and -image of a semi-metric space, where “semi-metric” can not be replaced by “metric”.
Let G be a compact group and X a G-ANR. Then X is a G-AR iff the H-fixed point set is homotopy trivial for each closed subgroup H ⊂ G.