Tent-maps, two-point sets, and the self-Tietze property
<p>This thesis discusses three distinct topics.</p> <p>A topological space <em>X</em> is said to be <em>self-Tietze</em> if for every closed <em>C ⊂ X</em>, every continuous <em>f : C &rightarrow; X</em> admits a continuous ex...
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| Format: | Thesis |
| Language: | English |
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2011
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| _version_ | 1826310191723839488 |
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| author | Davies, G |
| author2 | Knight, R |
| author_facet | Knight, R Davies, G |
| author_sort | Davies, G |
| collection | OXFORD |
| description | <p>This thesis discusses three distinct topics.</p> <p>A topological space <em>X</em> is said to be <em>self-Tietze</em> if for every closed <em>C ⊂ X</em>, every continuous <em>f : C &rightarrow; X</em> admits a continuous extension <em>F : X &rightarrow; X</em>. 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.</p> <p>A subset of the plane is a <em>two-point set</em> if it meets every straight line in exactly two points. We show that a two-point set cannot contain a dense <em>G<sub>δ</sub></em> 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.</p> <p>For ⋋ ∈ ℝ, the <em>tent-map with slope</em> ⋋ is the function <em>f</em> : [0, 1] &rightarrow; ℝ such that <em>f</em>(<em>x</em>) = ⋋<em>x</em> for <em>x</em> &leq; ½ and <em>f</em>(<em>x</em>) = ⋋(1 - <em>x</em>) for <em>x</em> &geq; ½. Properties of <em>w</em>-limit sets of tent-maps, i.e. sets of the form</p> <p> <table align="center" cellborder="0"> <tr> <td><span style="font-size: 250%;">⋂</span><br><em>n</em>∈ℕ</br></td> <td> _________________ <br>{ <em>fn</em> + <em>k</em> (<em>x</em>) | <em>k</em> ∈ ℕ }</br></td> </tr> </table> </p> <p>for <em>x</em> ∈ [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 <em>w</em>-limit set is given.</p> |
| first_indexed | 2024-03-07T07:48:27Z |
| format | Thesis |
| id | oxford-uuid:6aaa0726-062a-428c-8dbe-03754c4d5448 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:48:27Z |
| publishDate | 2011 |
| record_format | dspace |
| spelling | oxford-uuid:6aaa0726-062a-428c-8dbe-03754c4d54482023-06-28T11:51:08ZTent-maps, two-point sets, and the self-Tietze propertyThesishttp://purl.org/coar/resource_type/c_db06uuid:6aaa0726-062a-428c-8dbe-03754c4d5448Analytic Topology or TopologyDynamical systems and ergodic theory (mathematics)EnglishOxford University Research Archive - Valet2011Davies, GKnight, RCollins, P<p>This thesis discusses three distinct topics.</p> <p>A topological space <em>X</em> is said to be <em>self-Tietze</em> if for every closed <em>C ⊂ X</em>, every continuous <em>f : C &rightarrow; X</em> admits a continuous extension <em>F : X &rightarrow; X</em>. 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.</p> <p>A subset of the plane is a <em>two-point set</em> if it meets every straight line in exactly two points. We show that a two-point set cannot contain a dense <em>G<sub>δ</sub></em> 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.</p> <p>For ⋋ ∈ ℝ, the <em>tent-map with slope</em> ⋋ is the function <em>f</em> : [0, 1] &rightarrow; ℝ such that <em>f</em>(<em>x</em>) = ⋋<em>x</em> for <em>x</em> &leq; ½ and <em>f</em>(<em>x</em>) = ⋋(1 - <em>x</em>) for <em>x</em> &geq; ½. Properties of <em>w</em>-limit sets of tent-maps, i.e. sets of the form</p> <p> <table align="center" cellborder="0"> <tr> <td><span style="font-size: 250%;">⋂</span><br><em>n</em>∈ℕ</br></td> <td> _________________ <br>{ <em>fn</em> + <em>k</em> (<em>x</em>) | <em>k</em> ∈ ℕ }</br></td> </tr> </table> </p> <p>for <em>x</em> ∈ [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 <em>w</em>-limit set is given.</p> |
| spellingShingle | Analytic Topology or Topology Dynamical systems and ergodic theory (mathematics) Davies, G Tent-maps, two-point sets, and the self-Tietze property |
| title | Tent-maps, two-point sets, and the self-Tietze property |
| title_full | Tent-maps, two-point sets, and the self-Tietze property |
| title_fullStr | Tent-maps, two-point sets, and the self-Tietze property |
| title_full_unstemmed | Tent-maps, two-point sets, and the self-Tietze property |
| title_short | Tent-maps, two-point sets, and the self-Tietze property |
| title_sort | tent maps two point sets and the self tietze property |
| topic | Analytic Topology or Topology Dynamical systems and ergodic theory (mathematics) |
| work_keys_str_mv | AT daviesg tentmapstwopointsetsandtheselftietzeproperty |