By Krzysztof Maurin

The terribly speedy advances made in arithmetic in view that international conflict II have led to research turning into a tremendous organism unfold ing in all instructions. long gone for sturdy absolutely are the times of the nice French "courses of study" which embodied the complete of the "ana lytical" wisdom of the days in 3 volumes-as the classical paintings of Camille Jordan. probably for this reason present-day textbooks of anal ysis are disproportionately modest relative to the current state-of-the-art. extra: they've got "retreated" to the nation prior to Jordan and Goursat. in recent times the scene has been altering speedily: Jean Dieudon ne is delivering us his monumentel components d'Analyse (10 volumes) written within the spirit of the good French direction d'Analyse. To the simplest of my wisdom, the current publication is the single certainly one of its measurement: ranging from scratch-from rational numbers, to be precise-it is going directly to the speculation of distributions, direct integrals, research on com plex manifolds, Kahler manifolds, the idea of sheaves and vector bun dles, and so on. My goal has been to teach the younger reader the sweetness and wealth of the unsual global of contemporary mathematical research and to teach that it has its roots within the nice arithmetic of the nineteenth century and mathematical physics. i know that the younger brain eagerly beverages in appealing and tough issues, rejoicing within the proven fact that the area is superb and teeming with adventure.

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**Example text**

If the family of all compact subsets is taken for E, we obtain the concept of compact convergence (in the previous terminology, now obsolete, almost uniform convergence). 51 XII. TOPOLOGY 4. If X is also a uniform space, one may speak of precompact convergence: that is, E is a family of precompact sets. If X is a topological vector space, convergence on (pre)compact convex sets is then particularly important. 13. FAMILIES OF EQUICONTINUOUS MAPPINGS.

V E ~. e. the uniform structure U is induced by U). PROOF. (c) is proved exactly as in the case of a metric space (cf. 1). (d) is immediate in virtue of the observation that An (XxX) = Ji. it is an entourage in X constructed above. (a) The completeness of (X, U) will follow easily from a lemma below (we defer the proof until the proof of the theorem is finished). x 45 XII. 9. Let ~ be a uniformity basis of a uniform space (X, U). Then (X is complete) ¢> (every Cauchy net indexed by ~ has a limit in X.

For a topological space (X, or) the following conditions are equivalent: (a) (X, or) is a Baire space. 00 (b) For every open subset @ of X, if @ = U M i , then for at least ;=1 one index i the set Mi n @ contains an open nonempty subset of X. (c) For every countable family {A;}i:l of nowhere dense subsets of X U ;=1 AI is a boundary set in X. In other words, every set of the first category in X is a boundary set. (d) For every countable family {A i }r'=l of closed nowhere dense 00 subsets of X U ;=1 A; is a boundary set.