Quasiconformal mappings in the plane 2nd Edition (Die by Olli Lehto, K.I. Virtanen

By Olli Lehto, K.I. Virtanen

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The Cech homology groups of X are defined as the inverse limit ˇ q (X) := H lim ←− Hq (N (α)). Covf (X) If A is closed subset of X then every covering α ∈ Cov A can be obtained from a covering α ∈ Cov X satisfying Ui = A ∩ Ui , where Ui ∈ α. It is known that ˇ q (A) = lim Hq (N (α)). H ←− α ˇ q (A) using coverings in Covf (X) only. If B is Now, we would like to describe H a subset of X then by N (α)|B we denote the subcomplex of N (α) which consists of all simplexes σ with supp σ ⊂ B. 10) Proposition.

Vk+1 , . . 1) for p ≤ 3. After (k − 1) such steps we obtain the desired covering β. 10). 1). 9) states that Γ is a cofinal subfamily in Cov X. 1) ensures that the simplical complexes N (α) and N (α)|St(A,α) are simplically isomorphic. Therefore H∗(N (α)) = H∗(N (α)|St(A,α) ). Hence ˇ ∗ (A) = lim H∗ (N (α)) = lim H∗ (N (α)|St(A,α)) = lim H∗(N (α)|St(A,α) ) H ←− ←− ←− Γ and the proof is finished. Γ Cov X 34 CHAPTER I. BACKGROUND IN TOPOLOGY In the above the coefficient group was inessential. From now on we assume that the coefficient group is a field F .

X∈A In a similar way we obtain sup dist(z, A) ≤ dH (A, B) + dH (B, C). z∈C Consequently we obtain dH (A, C) ≤ dH (A, B) + dH (B, C) and the proof is completed. The metric dH defined on B(X) is called the Hausdorff distance or Hausdorff metric in B(X). 4) Theorem. (B(X), dH ) is a complete metric space whenever (X, d) is complete. Proof. Let {An } be a Cauchy sequence in B(X). We shall prove first that the set A defined as follows: ∞ A= ∞ cl n=1 Am m=n is nonempty, bounded and limn An = A. Let ε > 0 and N be the set of all natural numbers.

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