General Topology and Homotopy Theory by I.M. James

By I.M. James

Scholars of topology rightly whinge that a lot of the elemental fabric within the topic can't simply be present in the literature, no less than no longer in a handy shape. during this e-book i've got attempted to take a clean examine a few of this easy fabric and to arrange it in a coherent style. The textual content is as self-contained as i'll kind of make it and will be particularly available to a person who has an uncomplicated wisdom of point-set topology and staff idea. This publication is predicated on a process sixteen graduate lectures given at Oxford and in other places now and then. In a process that size one can't speak about too many subject matters with no being unduly superficial. even though, this was once by no means meant as a treatise at the topic yet particularly as a quick introductory path with the intention to, i'm hoping, end up precious to experts and non-specialists alike. The creation encompasses a description of the contents. No algebraic or differen­ tial topology is concerned, even if i've got borne in brain the desires of scholars of these branches of the topic. workouts for the reader are scattered in the course of the textual content, whereas feedback for additional examining are inside the lists of references on the finish of every bankruptcy. typically those lists contain the most assets i've got drawn on, yet this isn't the kind of ebook the place it truly is potential to provide a reference for every little thing.

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Mit den gegebenen Hilfsmitteln ist die Lösung der Aufgaben 5 nicht ganz einfach. Wir werden auf diese Probleme im Abschn. 2 über Symbolische Dynamik zurückkommen, der Hilfsmittel zu einer wesentlich einfacheren Behandlung dieser Fragen liefern wird (siehe Aufgabe 26). 3. Geben Sie insbesondere in einem dieser Beispiele Punkte an, die nichtwandernd und ebenso nicht rekurrent sind. Es seien (X, T) ein topologisches dynamisches System und Y ⊂ X eine nichtleere abgeschlossene T-invariante Teilmenge von X.

Dann gibt es eine Folge h n ∈ D mit ∥h n − f ∥ →  für n → ∞. Da ∫ h n d μ = ∫ h n dν für alle n gilt, folgt ebenso ∫ f d μ = ∫ f dν aus dominierter Konvergenz. 26 1 Topologische Dynamik Sei d eine Metrik auf X. Für jede offene Menge G ⊂ X und jedes n ≥  definieren wir die stetige Funktion g n (x) = min(nd(x, G c ), ) für x ∈ X. Es gilt lim n→∞ g n = G 6 und daher auch μ(G) = ν(G) wegen monotoner Konvergenz. Ebenso wegen monotoner Konvergenz folgt nun auch μ(⋂n G n ) = ν(⋂n G n ) für jede beliebige Folge von offenen Mengen.

Zeigen Sie, dass die Mengen der nichtwandernden Punkte von (X, T) und (Y , S) die Bedingung Ω(S) ⊂ Ω(T) ∩ Y erfüllen. Finden Sie ein Beispiel mit Ω(S) ≠ Ω(T) ∩ Y . (1) Zeigen Sie, dass die Expansivität eines dynamischen Systems nicht von der Wahl der Metrik d auf X abhängt. Wenn also T bezüglich einer Metrik d auf X expansiv ist, so ist es auch bezüglich jeder anderen Metrik d ′ auf X expansiv (natürlich muss auch d ′ die Topologie von X induzieren). Beim Wechsel der Metrik können sich aber die expansiven Konstanten ändern.

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