Download e-book for kindle: Les Fondements de la géométrie by Jacqueline Lelong-Ferrand

By Jacqueline Lelong-Ferrand

ISBN-10: 2130388515

ISBN-13: 9782130388517

Approche axiomatique de l. a. géométrie
* through los angeles constitution vectorielle
* comme constitution d'incidence (à l. a. façon de Veblen & younger)

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Extra resources for Les Fondements de la géométrie

Example text

Thus we IS] TOPOLOGY OF EUCLIDEAN PLANE 23 can apply Zorn’s lemma on d to get a maximal element A,, which is the maximal set having the property P. Hausdorffs lemma j Zermelo’s axiom. Let d be a collection of non-empty sets. We denote by 2 the set of the pairs (A, a ) where A is an element of d and a is an element of A. , for every (A, a ) E Y and (A’, a ’ )E Y , A = A’ implies a = a’. Then P is clearly a finite property. Therefore by use of Hausdorff’s lemma we can find a maximal subset Lf,, of Lf having the property P.

We mean by a set sequence a sequence A,, A,, . . of non-empty subsets of E 2 such that The set sequence {A,, 1 n = 1, 2 , . } is said to converge to a point p if for every E > 0 there exists an n for which A, C S , ( p ) . We take a point p,, E A,, from each member, A,,, of the set sequence. Then we obtain a point sequence {p,,} which is called a point sequence derived from {A,}. We can easily show that the set sequence {A,,}converges to p if and only if every point sequence derived from {A,} converges to p.

Nexandroff is highly recommended to the reader as a historical survey of the modern development in general topology. See his paper [ S ] too, for recent developments. 24 INTRODUCTION [IS As is well known, the concept of convergence of a point sequence is very significant in E’, especially in analysis of E 2 . We may say it is fundamental in the study of E 2 . This concept is closely related with such other concepts as neighborhood, closure, open set, etc. as seen in the following. e. S , ( p ) is often called the &-neighborhood of p.

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Les Fondements de la géométrie by Jacqueline Lelong-Ferrand


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