By Glen E. Bredon
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This e-book deals an introductory direction in algebraic topology. beginning with basic topology, it discusses differentiable manifolds, cohomology, items and duality, the basic staff, homology idea, and homotopy conception.
From the experiences: "An fascinating and unique graduate textual content in topology and geometry...a sturdy lecturer can use this article to create a great course....A starting graduate pupil can use this article to profit loads of mathematics."—-MATHEMATICAL REVIEWS
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Topology and Geometry (Graduate Texts in Mathematics, Volume 139)
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Theorem. Let X be a locally compact Hausdorff space. Put X + = Xu { OCJ} where OCJ just represents some point not in X. Define an open set in X+ to be either an open set in Xc X+ or X+ - C where C c X is compact. Then this defines a topology on X+ which makes X+ into a compact Hausdorff space called the "one-point compactification" of X. Moreover, this topology on X+ is the only topology making X+ a compact Hausdorff space with X as a subspace. The whole space X+ and 0 are clearly open. 5). The other cases of an intersection of two open sets are trivial.
A metric space X is totally bounded if, for each E > 0, X can be covered by a finite number of E-balls. 4. Theorem. In a metric space X the following conditions are equivalent: (I) X is compact. (2) Each sequence in X has a convergent subsequence. (3) X is complete and totally bounded. PROOF. (I) => (2) Let {xn} be a sequence. Suppose that X is not a limit of a subsequence. Then there is an open neighborhood U x of X containing Xn for only a finite number of n. Since X can be covered by a finite number of the U x' this contradicts the infinitude of indexes n.
This relies on knowing about the existence of sufficiently many, in some sense, continuous real valued functions on the space. That leaves open the question of finding purely topological assumptions that will guarantee such functions, and that is what we are going to address in this section. 1. Lemma. Suppose that, on a topological space X, we are given, for each dyadic rational number r = m/2n (0:::; m:::; 2n), an open set Ur such that r < s => Or C Us. Then the function f: X --+ R defined by ifxEU 1, if x¢U 1, is continuous.