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GENERAL TOPOLOGY - uni-bielefeldde

A MATHEMATICAL MISTAKE IN N. BOURBAKI'S

\GENERAL TOPOLOGY" My favorite books in General Topology are the books of N. Bour- baki. There are a lot of reasons for that. Among them I strongly believe that especially the part of Exercises is an endless source of deep results and a continuous inspiration for further research. But even in the legendary \General Topology" of N. Bourbaki there is at least one mistake! Precisely in the Exercise 13 of Ch. X, p. 323, part (d) in [1] it is said that ifEis a uniformly equicontinuous fam- ily of homeomorphisms of a locally compact uniform spaceXthen K(E) :=fx2X:Exis relatively compactgis a closed subset of X. This is not true ifEis not a subset of a uniformly equicontinuous groupof homeomorphisms ofXas we can easily see by the following counterexample.

Counterexample.Let

X=1[ k=1f(x;y) :x=1 k ; y¸0g [ f(x;y) :x= 0; y >0g be endowed with the Euclidean metric. Consider the familyE=ffng of selfmaps ofXde¯ned byfn(x;y) = (x;y n ). The familyEcon- sists of uniformly equicontinuous homeomorphisms ofXandK(E) =S1 k=1f(x;y) :x=1 k ; y¸0gas can be easily checked. Hence the set

K(E) is not closed inX.

References

[1] Bourbaki N.,Elements of Mathematics. General Topology, Parts I and II, Her- mann, Paris, 1966. 1quotesdbs_dbs2.pdfusesText_2