By N. Bourbaki
This is the softcover reprint of the English translation of 1971 (available from Springer considering that 1989) of the 1st four chapters of Bourbaki's Topologie générale. It offers the entire fundamentals of the topic, ranging from definitions. very important sessions of topological areas are studied, uniform constructions are brought and utilized to topological teams. actual numbers are built and their homes demonstrated. half II, comprising the later chapters, Ch. 5-10, can be on hand in English in softcover.
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Extra resources for General Topology: Chapters 1–4
For V contains UnA, where U is an open subset of X which contains x, hence V contains UnA = U (§ I, no. 6, Proposition 5). PROPOSITION 3. Let (A')'EI be afamily of subsets of a topological space X, such that one of the following properties holds: a) The interiors of the A, cover X. b) (A')'EI is a locally finite closed covering of X (§ I, no. 5). Under these conditions, a subset B of X is open Crespo closed) in X only if each of the sets B n ~ is open (resp. closed) in AI' if and Clearly if B is open (resp.
B) The image under f of every open set which is saturated with respect to R is an open set in the subspace f (X). c) The image under f of every closed set which is saturated with respect to R is a closed set in the subspace f (X) . For the condition b) [resp. c)] expresses that the image under g of every open (resp. closed) set in X/R is an open (resp. closed) set in f(X). Example. eJ a covering of X, Y the sum of the subspaces X. : Y. - X.. f. on Y. for each ~ e I, and let R be the equivalence relation f (x) = j (y) ; the quotient space Y/R is thus obtained by" pasting together" the Y.
Remark. PROPOSITION 5. For a subset L of a topological space X, tke following properties are equivalent: a) L is locally closed in X. UOTmNT SPACES b) L is the intersection of an open subset and a closed subset of X. c) L is open in its closure L in X. If L is locally closed, then, for each x e L, there is an open neighbourhood = U Va; is open a;eL in X, and Proposition 3 of no. I shows that L is closed in U; therefore a) implies b). If L = Un F, where U is open and F closed in X, we have L c: F j hence L c: U n L c: U n F = L, which shows that L = Un L is open in L, so that b) implies c).