Get Algebraic Topology: An Intuitive Approach PDF

By Hajime Sato

The only so much tricky factor one faces whilst one starts off to benefit a brand new department of arithmetic is to get a believe for the mathematical experience of the topic. the aim of this ebook is to aid the aspiring reader gather this crucial logic approximately algebraic topology in a quick time period. To this finish, Sato leads the reader via uncomplicated yet significant examples in concrete phrases. furthermore, effects aren't mentioned of their maximum attainable generality, yet when it comes to the easiest and so much crucial situations.

In reaction to feedback from readers of the unique version of this booklet, Sato has additional an appendix of worthwhile definitions and effects on units, basic topology, teams and such. He has additionally supplied references.

Topics coated contain basic notions resembling homeomorphisms, homotopy equivalence, basic teams and better homotopy teams, homology and cohomology, fiber bundles, spectral sequences and attribute periods. items and examples thought of within the textual content contain the torus, the Möbius strip, the Klein bottle, closed surfaces, phone complexes and vector bundles.

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Insbesondere ist das Bild eines Intervalls I unter einer stetigen Funktion I → R wieder ein Intervall: Der Zwischenwertsatz der Analysis erscheint hier als Spezialfall. 3 Die Vereinigung einer Familie zusammenhängender Unterräume eines topologischen Raumes, welche paarweise nicht disjunkt sind, ist selbst zusammenhängend. Beweis: Seien (Zi | i ∈ I) die Familie von Unterräumen und Zi −→ S 0 f: i∈I eine stetige Abbildung, so ist deren Einschränkung auf jeden Unterraum Ui konstant. Weil die paarweisen Durchschnitte nicht leer sind, muss also f konstant sein.

Eine offene Überdeckung heißt endlich oder abzählbar, wenn dies auf die Indexmenge zutrifft. Hat jede offene Überdeckung von X eine endliche Teilüberdeckung, so wird X kompakt genannt. Kompaktheit ist offenbar eine topologische Invariante, sind zwei Räume also homöomorph zueinander, so ist der eine kompakt genau dann, wenn der andere es ist. Beispiel: Das Intervall [0, 1] ist kompakt, auch wenn das mit der eben gegebenen Definition nicht offensichtlich ist und deswegen auch begründet werden soll. Sei dazu (Uj | j ∈ J) eine offene Überdeckung von [0, 1].

7 (Tietze-Urysohn) Für jeden topologischen Raum X sind äquivalent: (T4) Der Raum X erfüllt T4. (UE) Zu je zwei disjunkten abgeschlossenen Teilmengen A, B von X existiert eine Urysohn-Funktion. (TE) Zu A ⊆ X abgeschlossen und f : A → R stetig gibt es eine stetige Abbildung F : X → R, die mit f auf A übereinstimmt. Beweis: In T4-Räumen kann man zu je zwei abgeschlossenen disjunkten A, B eine Menge C finden, die ◦ A ⊆ C ⊆ C¯ ⊆ X \ B erfüllt. Hierzu muss man nur A und B durch disjunkte Umgebungen U, V trennen und U = C wählen.

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