By Heikki Junnila
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Ce cours de topologie a été dispensé en licence à l'Université de Rennes 1 de 1999 à 2002. Toutes les constructions permettant de parler de limite et de continuité sont d'abord dégagées, puis l'utilité de l. a. compacité pour ramener des problèmes de complexité infinie à l'étude d'un nombre fini de cas est explicitée.
This ebook is the 6th version of the vintage areas of continuous Curvature, first released in 1967, with the former (fifth) variation released in 1984. It illustrates the excessive measure of interaction among team concept and geometry. The reader will enjoy the very concise remedies of riemannian and pseudo-riemannian manifolds and their curvatures, of the illustration concept of finite teams, and of symptoms of modern growth in discrete subgroups of Lie teams.
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Extra info for A second course in general topology
By Lemma 4, there exists a continuous mapping fn : X → Y such that we have d(fn (x), θ(x)) < 2−n−1 for every x ∈ X. The definition of θ shows that fn satisfies both required conditions. This completes the induction. By completeness of Y , there exists a function f : X → Y such that fn (x) → f (x) for every x ∈ X. The convergence is uniform, and hence the mapping f is continuous. For every x ∈ X, we have that d(f (x), ϕ(x)) = 0, and it follows, since ϕ(x) ⊂c Y , that f (x) ∈ ϕ(x). As a consequence, f is a continuous selection of ϕ.
We can define a mapping ϕ : X → Sℓ2A by the formula ϕ(x)α = fα (x). We show that ϕ is an embedding. Since X is T1 and F induces the topology of X, we see that ϕ is one-to-one. To show that ϕ is continuous, let xn → x in X. We show that ϕ(xn ) → ϕ(x), in other words, that √ ||ϕ(xn ) − ϕ(x)|| → 0. For all a, b ≥ 0 we have that ab ≤ 12 (a + b), and it follows that we have, for every n ∈ N that α∈A fα (xn )fα (x) ≤ 1 2 fα (xn ) + α∈A and hence that 49 α∈A fα (x) = 1 < ∞ ||ϕ(xn ) − ϕ(x)||2 = fα (xn ) + α∈A α∈A fα (x) − α∈A fα (xn ) − α∈A fα (x) 2 fα (xn )fα (x) = 2 − 2 2 = fα (xn )fα (x) .
Proof. Let g : X By Proposition 11, f is continuous, and it follows by Lemma 17, that f is Z-uniformly ˇ continuous. By Lemma 16, there exists an Z-uniformly continuous function fˇ : UZ → R such that we have fˇ(Zx ) = f (x) for every x ∈ X. The function fˇ is an extension of g and Lemma 15 shows that fˇ is continuous UZ → I. ˜ and we have another representation of the Cech-Stone ˇ We can thus write UZ = β X, compactification of X. , with discrete spaces. 19 Example Let D be a discrete space. Then every subset of D is a zero-set, that is, ZD = P(D).