CHAPTER 1
The Ramsey-Dvoretzky-Milma n phenomeno n
1.1. Finit e oscillatio n stabilit y
Uniform rathe r tha n metri c space s provid e th e mos t natura l settin g fo r
studying th e Ramsey-Dvoretzky-Milma n phenomenon .
DEFINITION
1.1.1Recal . l that a uniform space is a pair (X,U), consistin g o f a
set X an d a uniform structure, U, o n X , tha t is , a collectio n o f subset s o f X x X
(binary relation s o n X) , calle d entourages of the diagonal, satisfyin g th e followin g
properties.
(1) Th e famil y U i s close d unde r finite intersection s an d superset s (i f V G ti
a n d V C l / C I x I , the n U G U).
(2) Ever y V G hi contains th e diagona l A = {(x , x): x G X}.
(3) I f V G U, the n V' 1 = {(x, y) : (y, x) G V} i s in U.
(4) I f V G W, there exists &U eU suc h that E/ o J7 = {(re , 2;): 3y G X, (x , y) G
(7, (y , 2:) G U} i s a subset o f V.
DEFINITION
1.1.2A . subfamil y B CU is called a basis of th e uniformit y U if
for ever y U,V £ B ther e i s a C G B wit h C C ^ 4 Pi ^, an d ever y entourag e F e W
contains, a s a subset, a n elemen t U G 25 .
A family2 3 of subsets o f a set X serve s as a basis for a uniform structur e i f an d
only i f it satisfie s th e followin g conditions :
(1) Fo r ever y V , [/" G B ther e i s a C G B wit h C C AnB (tha t is , B form s a
prefilter).
(2) Ever y V E B contain s th e diagona l A .
(3) I f V G 23, then fo r som e J 7 G 23, U C F " 1 .
(4) I f y G 23, there exist s a 17 G 23 such tha t 17 o 17 C V.
For ever y elemen t y o f th e unifor m structur e U an d eac h x G X , denot e
y[x] = { y G X: (x,y) G V}. Thi s i s th e V-neighbourhood o f # . Th e set s V[x] ,
where 7 G^ , for m a neighbourhoo d basi s fo r x wit h regar d t o a certai n topolog y
on X , calle d th e topolog y determined by , o r associated to , Z^ . I f Z Y is a unifor m
structure determinin g th e topolog y o f a give n topologica l spac e X , the n U i s sai d
to b e compatible.
\iU i s separated (a s we will usually assume) , tha t is , P\U = A , the n th e associ -
ated topolog y i s Tychonoff. Ther e i s a converse statemen t a s well.
EXERCISE
1.1.3 . Prov e tha t ever y Tychonof f topologica l spac e admit s a com -
patible unifor m structure , an d tha t ever y suc h structur e i s necessaril y separated .
(Hint: i f / i s a real-value d functio n o n a set, the n {(# , y): \f(x) f(y)\ s} i s a n
entourage o f the diagona l i n X x X.)
11
http://dx.doi.org/10.1090/ulect/040/02
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