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Title: Tent-maps, two-point sets, and the self-Tietze property
Author: Davies, Gareth
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2011
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This thesis discusses three distinct topics. A topological space X is said to be self- Tietze if for every closed C eX, 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, zero- dimensional, 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 G8 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 A E lR, the tent-map with slope A is the function f: [0, 1] -+ lR such that f(x) = AX for x :=:; ~ and f(x) = A(l - x) for x ~ ~. Properties of w-limit sets of tent-maps, i.e. sets of the form n {fn+k(x) I kEN} nEN for x E [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.
Supervisor: Knight, Robin ; Collins, Peter Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Analytic Topology or Topology ; Dynamical systems and ergodic theory (mathematics) ; tent-map ; two-point set ; self-Tietze ; retractifiable