Algebraische Topologie by Prof. Dr. Karl Heinz Mayer (auth.)

By Prof. Dr. Karl Heinz Mayer (auth.)

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Example text

Zeigen Sie, daB F = {U c IR n I/Rn \ U ist endlich oder 0 E IR n \ U} eine Topologie auf IR n ist und (/Rn ,F) normal ist. 5. Es seien X ein T2 -Raum, Reine Aquivalenzrelation auf X und 1r : X -+ XI R die natiirliche Projektion. Zeigen Sie: Wenn eine stetige Abbildung s : XI R -+ X existiert mit 1r 0 S = Id, so ist XI R hausdorffsch. 6. X und Y seien topologische lliiume, Y hausdorffsch und I, g : X -+ Y seien stetige Abbildungen, die auf einer dichten Teilmenge von X iibereinstimmen. Folgern Sie, daB f = gist.

20 Bemerkungen. Die Teilraumtopologie wurde von F. Hausdorff in seinem grundlegenden Buch 1914 angegeben. H. Tietze definierte 1923 Topologien fur das Produkt und fiir die Summe von topologischen lliiumen. 4 vorgestellten iiberein. Eine Basis fiir die Produkttopologie bilden bei Tietze alle Produkte von offenen Mengen in den einzelnen Faktoren. 4 definierte und nach dem in der Einleitung formulierten Prinzip "natiirliche" Topologie wurde von A. Tychonoff 1930 angegeben. Eine Definition der Quotiententopologie findet sich bei P.

Da h- 1 (W) offen ist, ist Ao offen. 23 eine kompakte Umgebung C von ao. Dann sei § 4. Kompakte Riiume 49 U = {y E X I {y} x C C h-l(W)}. Es wird gezeigt, daB U offen ist und daB U = j-l j(U) ist. Damit ist auch j(U) offen in Y, und j(U) x C ist eine Umgebung von (xo,ao) mit j(U) xC c W. Daher ist W Umgebung jedes seiner Elemente und damit offen. U ist offen: Wenn y E U ist, dann ist {y} x C C h-l(W). ), also V c U. j-l j(U) = U: Zunachst ist U C j-l j(U). Andererseits ist j-l j(U) X C = h- 1 h(U X C) c h- 1 (h h- 1 (W)) = h-l(W).

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