By Everitt
Complaints of a world convention prepared by means of the London Mathematical Society, held July 1987 on the U. of Birmingham, and ruled through the ghosts of Hardy, Littlewood and Polya, whose Inequalities (still the first reference within the box) seemed in 1934. 13 essays summarize subse
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A topological space X is called locally compact if for each x E X, there is an open set V such that x E V and Vis compact. 54. Every compact space is locally compact; Rn(n > I) (with usual topology) are locally compact, but not compact. 55. Let X be a locally compact Hausdorff space. We can compactify X by considering X* = X u {oo} and taking a set in X* to be open if and only if it is either an open subset of X or the complement of a compact subset of X. Clearly, the topological space X* (thus formed) is a compact Hausdorff space.
Letting (B;)s be the collection in 9 given by 91, Ui'= 1A; Bt =At, B2 = (A, u A 2 ) - A 1 , Ba = (A 1 U A 2 u A 3 ) - (A, u A2 ), we have a pairwise disjoint sequence in 9 and u~,A; = u~,B;. 12. If >-J" is a class of subsets of X for which A • whenever A and Bare in ~f, then I Proof. Clearly 9(>-r') C a(~r') as a(g") is a Dynkin system. To show a(ro:f) C :/:(r·t') it suffices to show that 9(,:)) is a a-algebra. 9 (if) whenever A and B are in 9(8"). t::4 ={BE 2X: An BE9(7S')}. t:: 1 is a Dynkin system.
The set of all interior points of A is called its interior and denoted by A 0 • I Remarks. 23. 24. 25. Let (X, ~) be a topological space and A, B C X. 19. E r;;. 0. The boundary Ab of A is the set of all its boundary points. 26. 27. 28. = I A- A 0 • (A U Bh CAb U Bb. 4)b = Ab. 20. A mapping f: X~ Y where (X, ~1 ) and (Y, iF2 ) are topological spaces, is called continuous (relative to iF1 and iF2 ) if J- 1 ( V) E iF1 for each V E iF2 • The mapping f is called open (or closed, respectively) if f(T) is open (closed) for each open (closed) T.