Tent-maps, two-point sets, and the self-Tietze property

Thesis

This thesis discusses three distinct topics.

A topological space X is said to be self-Tietze if for every closed C ⊂ X, every continuous f : C → X admits a continuous extension F : X → X. We show that every disconnected, self-Tietze space is ultranormal. The Tychonoff Plank is an example of a compact self-Tietze space which is not completely normal, and we establish that a completely normal, zerodimensional, homogeneous space need not be self-Tietze.

A subset of the plane is a two-point set if it meets every straight line in exactly two points. We show that a two-point set cannot contain a dense G_δ subset of an arc. We also show that the complement of a two-point set is necessarily path-connected. Finally, we construct a zero-dimensional subset of the plane of which the complement is simply-connected.

For ⋋ ∈ ℝ, the tent-map with slope ⋋ is the function f : [0, 1] → ℝ such that f(x) = ⋋x for x ≤ ½ and f(x) = ⋋(1 - x) for x ≥ ½. Properties of w-limit sets of tent-maps, i.e. sets of the form

⋂ n∈ℕ

_________________ { fn + k (x) | k ∈ ℕ }

for x ∈ [0, 1], are examined, and an example of a tent-map and a closed, invariant, nonempty, internally chain transitive subset of [0, 1] which is not an w-limit set is given.

Authors: Davies, G

• Publication date: 2011
• DOI: 10.5287/ora-mzyrk52zn
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: self-Tietze; tent-map; retractifiable; two-point set
• Subjects: Dynamical systems and ergodic theory (mathematics); Analytic Topology or Topology