Connection of Conservation Laws with the Geometry of

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For p F BT-T, let U(p) denote the collection of closed neighborhoods of p. Since = {p}, BT being Hausdorff, then T C c(nu(p)) u = cu. UFU (PI If T is Lindelaf (CU) ucu (P) possesses a countable subcover (CUn or, n T = @. Thus p 4 UT. V equivalently, (nun) Though completely regular Hausdorff Lindel6f spaces are replete, the converse is false. To see this, let T denote with the topology generated by the half-open intervals (a,bl. T is completely regular, Hausdorff and LindelEf (see below), hence replete, hence so is TXT.

F-') ,f (x), E ) of 0 (p*-f-') = (a -f'*o;(p) c US. v As sets, C(T,R) and C (T,R) = C(BT,R) obviously differ if and only if b T is not pseudocompact. But, tho different, could they be isomorphic for non-pseudocompact T? If so, then UT = U(BT) = BT -hence T is pseudocompact. 6-3) is Banach's result (Banach 1932, p. 170, Theorem 3 ) that in the class of compact metric spaces S and T, S and T are 32 1. ALGEBRAS OF CONTINUOUS FUNCTIONS homeomorphic if and only if C(S,R) is isometric to C(T,R) when each space carries its sup norm metric.

J . J To prove the inequality for 0 < x < 1, note first that for each k T = U x-1([(i-l)/k,i/k)). 7 Letting Z. MEASURES, BT AND ULAM CARDINAIS 37 -1 he the zero set x ([i/k,m)), this becomes k T = U (Zi-l-Zi) i=l x defined at t Now the simple functions 5 and x(t) = (i-l)/k satisfy the inequality 5 and 5x5 x. - Zi by Q Zi-l x(t) = i/k (i=l,. ,k) Thus and k c l/k m(Zi) = i=l As lim sup m ( 2 . ) for each i, it follows that, taking superior k k limits in (1) and replacing (l/k) C lim sup m (Zi) by (l/k) C m(Zi), P U i=l i=l and using (2).

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