Algebraische Topologie - download pdf or read online

By Wolfgang Lück

ISBN-10: 3528032189

ISBN-13: 9783528032180

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By setting := U1 ∩ · · · ∩ Un , this amounts to saying that ⊂ V ∪ W . As V and W are disjoint, we deduce that ∩ V = \W . Consequently, the set ∩ V is a clopen neighborhood of x in X . It follows that Q x ⊂ ∩ V . Therefore we have that Q x = A. This shows that Q x is connected. 8 Let X be a locally compact Hausdorff space. Then the following conditions are equivalent: (a) X is scattered; (b) X is totally separated; (c) X is totally disconnected. 6. 5 shows that (b) implies (c). Suppose that X is totally disconnected.

Fix a point x0 ∈ X and a sequence (an )n∈N of points of X such that the set {an | n ∈ N} is dense in X . Show that the sequence (d(x, an ) − d(x0 , an ))n∈N is in ∞ (R) for every x ∈ X and that the map ϕ : X → ∞ (R) defined by ϕ(x) = (d(x, an ) − d(x0 , an ))n∈N is an isometric embedding. 15 Show that the Banach space ∞ (R) is not separable. 16 Show that every second-countable scattered accessible space is homeomorphic to a subset of the Cantor set. 17 A metric space (X, d) is called an ultrametric space if one has d(x, y) ≤ max(d(x, z), d(y, z)) for all x, y, z ∈ X .

A subset of a Lindelöf space is not necessarily Lindelöf (see the example in Sect. 4). However, we have the following result. 6 Every closed subset of a Lindelöf space is itself Lindelöf. Proof Let X be a Lindelöf space and F a closed subset of X . Let α = (Ui )i∈I be an open cover of F. Then we can find, for each i ∈ I , an open subset Vi of X such that Ui = Vi ∩ F. As the family (Vi )i∈I ∪ {X \F} is an open cover of X and X is Lindelöf, there exists a countable subset J ⊂ I such that the family (V j ) j∈J ∪{X \F} covers X .

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Algebraische Topologie by Wolfgang Lück


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