Boundedly Controlled Topology. Foundations of Algebraic by Douglas R. Anderson, Hans J. Munkholm

By Douglas R. Anderson, Hans J. Munkholm

A number of fresh investigations have targeted recognition on areas and manifolds that are non-compact yet the place the issues studied have a few type of "control close to infinity". This monograph introduces the class of areas which are "boundedly managed" over the (usually non-compact) metric area Z. It units out to increase the algebraic and geometric instruments had to formulate and to turn out boundedly managed analogues of some of the ordinary result of algebraic topology and straightforward homotopy thought. one of many topics of the ebook is to teach that during many circumstances the evidence of a regular consequence might be simply tailored to end up the boundedly managed analogue and to supply the main points, usually passed over in different remedies, of this model. for that reason, the ebook doesn't require of the reader an in depth history. within the final bankruptcy it's proven that certain circumstances of the boundedly managed Whitehead team are strongly with regards to reduce K-theoretic teams, and the boundedly managed conception is in comparison to Siebenmann's right uncomplicated homotopy conception whilst Z = IR or IR2.

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Extra resources for Boundedly Controlled Topology. Foundations of Algebraic Topology and Simple Homotopy Theory

Example text

Note that the Hausdorff distance dH can be extended onto B(E) but it is no longer a metric. In fact, it is easy to see that dH (A, cl A) = 0. We shall define the measure of noncompactness on B(E). We shall say that a subset A ⊂ E is relatively compact provided the set cl A is compact. 9) Definition. Let E be a Banach space and B(E) the family of all bounded subsets of E. Then the function: α: B(E) → R+ defined by: α(A) = inf{ε > 0 | A admits a finite cover by sets of diameter ≤ ε} is called the (Kuratowski) measure of noncompactness, the α-MNC for short.

Let r: E → cl B(0, 1) be the retraction map defined as follows: x if x ≤ 1, r(x) = x if x > 1. x Let A ∈ B(E). Since r(A) ⊂ conv(A ∪ {0}), we obtain γ(r(A)) ≤ γ(A). In other words we can say that r is a nonexpansive map with respect to the Kuratowski or Hausdorff measure of noncompactness. Finally, note that the following version of the Cantor theorem holds true. 14) Theorem. If γ = α or γ = β and {An } is a decreasing sequence of ∞ closed nonempty subsets in B(E) such that limn γ(An ) = 0. Then A = n=1 An is a nonempty and compact subset of E.

The same correction is done for every triple of sets Vi , Vj , Vl (1 ≤ i < j < l ≤ k). Taking the intersections of such Vi , one obtains a covering α2 = {V1 , . . , 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.

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