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PREFACE
This book is intended as a text for the uniform space , a branch of topology,
there are no formal subject matter pre requisites for studying most of this book. I
do not even assume the reader known much set theory. Having said that , I must
hasten to add that unless the reader has studied a bit of analysis or rigorous
calculus, he will be missing much of the motivation for the concepts introduced
in this book. Things will go more smoothly if he Already has some experience
with continuous functions, open and closed sets , metric spaces and the like
although none of these is actually assumed. Most students in topology course
have , in my experience, some knowledge of the foundations of mathematics.
But the amount varies a great deal from one student to another. Therefore I begin
with a fairly through chapter on introduction of some important definitions that
is useful to study uniform space . It treats those topics which will be needed
later in the book.
CONTENT
TITLE PAGE NO.
UNIFORMITIES AND BASIC DEFINITIONS 1
METRISATION 10
COMPLETENESS AND COMPACTNESS 20
BIBLIOGRAPHY 32
2
DEFINITION 1.1
Let (X;d) be a metric space. Then a subset E of ๐ ร ๐ is said to be
an entourage if there exists ํ>0 such that for all ๐ฅ, ๐ฆ โ ๐, ๐ ๐ฅ, ๐ฆ <
ํ implies ๐ฅ, ๐ฆ โ E.
REMARK 1.2
Given a matric d on X and ํ> 0 , let Uฮต denote the set ๐ฅ, ๐ฆ โ
๐ ร ๐/๐(๐ฅ, ๐ฆ) < ํ . If X is the topology introduced by d and ๐ ร ๐ the
product topology , then uฮต is an open neighborhood of the diagonal โ๐ฅ in
๐ ร ๐. Thus every entourage is a neighbourhood of โ๐ฅ in ๐ ร ๐.
RESULT 1.3
Let ๐; ๐ , ๐; ๐ be matric spaces. Let ๐: ๐ โ ๐ be a function. Then
๐ is uniformly continuous (with respect to ๐ and ๐) if and only if for every
entourage ๐น of ๐; ๐ , there exist an entourage ๐ธ of ๐; ๐ such that for all
๐ฅ, ๐ฆ โ ๐ธ , ๐ ๐ฅ , ๐(y ) โ ๐น.
RESULT 1.4
Let {๐ฅn} be a sequence in a matric space ๐; ๐ . Then {๐ฅn} is a
Cauchy sequence if and only if for every entourage E there exist pโlN such
that for all m,nโฅ p, (๐ฅm,๐ฅn) โ E
RESULT 1.5
Let ๐ be any set and {๐n : ๐ โ ๐} nโฌN be a sequence of functions into
a matric space ๐; ๐ . Then {xn} converges uniformly to a function ๐: ๐ โ ๐
if and only if for every entourage , there exist mโN such that for all nโฅm
and ๐ฅ โ ๐ ๐n ๐ฅ , ๐ ๐ฅ โ E
3
PROPOSITION 1.6
Let ๐; ๐ be a pseudo-metric space and U the family of all its
entourages. Then
(i) if โ๐ฅ โ U for each U โU
(ii) if U โU then U-1โU
(iii) if UโU then there exist a Vโ U such that VโฆVโU
(iv) if U,Vโ U then UโฉVโ U
(v) if U โ U and Uโ V โ ๐ ร ๐ then Vโ U
If, moreover d is a matric then in addition to the above, we also have
(vi) โฉ{ U /U โ U } =โ๐ฅ={( ๐ฅ, ๐ฅ)/๐ฅ โ ๐}
Proof
(i)
given ๐; ๐ is a pseudometric
then ๐(๐ฅ,๐ฅ)=0 for all ๐ฅ โ ๐
โ U is reflexive
We know that U is reflexive if and only if โ๐ฅ โU
โ โ๐ฅ โ U for each U โ U
(ii)
since d is a pseudometric d(๐ฅ,๐ฆ)= d(y, ๐ฅ)
โ U is symmetric
โ U = U -1
โ U -1โ U
4
(iii)
suppose U โ U then there exist ํ >0 such that Uํ โ U
Where Uํ={(๐ฅ,y)โ ๐ ร ๐/ d(x,y)< ํ}, let V=Uํ/2
Now UโฆV ={(๐ฅ,y)โ ๐ ร ๐/ โ zโ ๐ such that (๐ฅ,z)โ V, (z,y)โU}
To prove VโฆVโU , let (๐ฅ,y)โ VโฆV.
โthere exist z โ ๐ such that (๐ฅ,z)โV, (y,z)โV
โd(๐ฅ,z)< ํ/2 and d(y,z)< ํ/2
โd(๐ฅ, y)< ํ
โ (x,y)โU
โด VโฆVโ U
(iv)
let ๐ฟ,ํ be such that Uฮด โ V and Uํ โ U
let ๐ผ=min(๐ฟ,ํ) then Uฮฑโ UโฉV
there exist (๐ฅ,y)โ ๐ ร ๐ such that d(x,y)< ๐ผ
โ (๐ฅ,y)โU and (x,y)โV
โ( ๐ฅ,y)โ UโฉV
โ UโฉV is an entourage
โ UโฉV โ U
(v)
given Uโ U and Uโ V โ ๐ ร ๐
5
Since Uโ U , there exist ํ > 0 such that d(x,y)< ํ โ (x,y)โU
โ (x,y)โV since UโV
ie, there exist ํ > 0 such that d(x,y)< ํ โ (x,y)โ V
โ V is an entourage
โ Vโ U
(vi)
given d is a metric,
ie, d(x,y)> 0 for all ๐ฅ โ ๐ฆ
โโฉ{U:Uโ U } =โ๐ฅ
DEFINITION 1.7
A uniformity on a set X is a nonempty collection U of ๐ ร ๐
Satisfying the following properties.
(i) โ๐ฅ โU for each Uโ U
(ii) if U โ U then U-1โ U
(iii) if U โ U then there exist Vโ U such that VโฆVโ U
(iv) if U,Vโ U , then Uโฉ V โ U
(v) if U โ U and U โ V โ ๐ ร ๐ then Vโ U
Members of U are called entourages. The pair (X, U ) is called uniform
space.
6
DEFINITION 1.8
A uniform space (X; U ) is said to be Hausdorff or separated if for a
metric d, โฉ {๐: UโU}=โ๐ฅ. Then U is called Hausdorff uniformity.
DEFINITION 1.9
A uniform space (X; U ) is said to be pseudo-metrisable ( or
metrisable) if there exist a pseudo-metric (respectively a metric) ๐ on X
such that U is precisely the collection of all entourages of (X;๐) in such a
case we also say that U is the uniformity induced or determined by ๐ .
DEFINITION 1.10
Let (X; U ) be a uniform space. Then the subfamily B of U is said to
be a base for U if every member of U contains some members of B ; while
a subfamily S of U is said to be sub-base for U if the family of all finite
intersections of members of S is a base for U.
PROPOSITION 1.11
Let X be a set and SโP ๐ ร ๐ ) be a family such that for every
UโS the fol lowing conditions hold:
a) โ๐ฅ โ U
b) U-1
contains a member of S , and
c) there exists VโS such that VโฆVโU, then there exists a unique
uniformity U for which S is a sub-base.
Proof
Let B be the family of all finite intersections of members of S and
7
let U be the family of all supersets of members of B. We have to prove
that U is a uniformity on X.
From condition a) โx โ U
Let UโU and Uโ ๐ โ ๐ ร ๐
Here UโS. from the construction of U, it is clear that VโU Now we prove that U
-1โU, for this suppose UโU, then there exist
U1,U2,โฆUnโS such that ๐๐๐=1 iโ U
By b) each U
-1contains some ๐iโS
Then V๐
๐=1 i โ Uni=1 i
-1 โ U-1
So ๐๐
๐=1 iโB and U-1โU
Now we prove that if UโU then there exist a VโU such that VโฆVโU
Suppose UโU and U 1, U2,โฆ. U nโS such that ๐๐๐=1 iโU
Now by c) each i=1,2,โฆ.n we can find Vi โS such that
ViโฆV iโ Ui
Let V = ๐๐
๐=1 i
Then VโฆV โ (๐๐
๐=1 iโฆ Vi)
ie, VโฆV โ (๐๐
๐=1 iโฆ Vi) โ U๐๐=1 i โU
ie, VโฆV โ U
further VโB and hence VโU Now we prove that if U,VโU then UโฉV โU
let U,Vโ U , we can find U1, U2,โฆ. Un and V1, V 2,โฆ V n โ S such that
V๐๐=1 j โV
8
then ( U๐๐=1 i)โฉ( V๐
๐=1 j) โB .
Since UโฉV is a superset of this intersection , it follows that UโฉVโ U.
Thus U is a uniformity for X. But its very construction B is a base and
S is a subbase for U.
Uniqueness of U is trivial since a sub-base determines uniformity.
THEOREM 1.12
For a uniform space (X; U) , let Iu be the family {Gโ X: for each xโ
G, โ U โU, such that U[x]โG}. Then Iu is a topology on X.
Proof
Clearly X,โ โ Iu .
Also from the very nature of the definition, it is clear that Iu is
closed under arbitrary unions.
Now we show that Iu is closed under finite intersections.
Let G , H โ Iu and suppose xโ ๐ โฉH.
Then there exist U,VโU such that U[x] โ G and V[x] โ H
Let W=UโฉV, then WโU Also W[x] โ U[x]โชV[x]
So W[x] โ GโฉH
so GโฉHโ Iu , thus Iu is a topology on X.
9
DEFINITION 1.13
Given a uniform space (X;U ) and a subset Y of X, we defined the
relativised uniformity U/Y as the family {Uโฉ(YรY):UโU} here (Y; U/Y)
is said to be a uniform subspace of (X;U ).
PROPOSITION 1.14
Let (X;U ), (Y;V ) be uniform spaces and f:XโY a function which is
uniformly continuous with respect to U and V .let Iu and Iv be the
topology on X and Y respectively as given by above theorem. Then f is
continuous with respect to these topologies.
Proof
Let Gโ Iv. we have to show that f -1(G) is open in X.
ie, f -1(G)โIu. for this let xโ f -1(G), then f(x)โG.
by the definition of Iv there exist VโV such that V[f(x)]โG.
Let U={(z,y)โ X รX : (f(z), f(y))โV },
then UโU , since f is uniformly continuous.
Moreover, U[x] โ f -1(V[f(x)]) โ f -1(G).
So f -1(G) is open in X. Hence f is continuous.
DEFINITION 1.15
Let (X;U ) be a uniform space. Then the topology Iv is called a
uniform topology on X induced by U. A topological space (X,I) is said to
be uniformisable if there exist a uniformity U on X such that I=Iu
11
PROPOSITION 2.1
Suppose a uniformity U on a set ๐ has a countable base. Then
there exists a countable base {Un}โ
n = 1 for U such that each Un is
symmetric and Un+1โฆUn+1โฆ Un+1โUn for all n โN
Proof:
Let the given countable base be V = {V1, V2, V3โฆ. Vnโฆ}. Set U1=V1โฉ
V1-1
Then U1 is a symmetric member of U .We know that for every member
U of U, there exists a symmetric member V of U such that VโฆVโฆVโU.
Consider the set U1โฉV2 .
Here U1โฉV2โU .Then there exists a symmetric member say U2 of U
such that
U2โฆU2โฆU2 โ U1โฉV2 .Next consider U2โฉ V3 .
Then we get a symmetric member U3 of U such that
U3โฆ U3โฆ U3 โ U2โฉV3.
In this manner, we proceed by induction and get a sequence
{Un:nโN} of symmetric members of U such that for each nโN
,
Un+1โฆUn+1โฆUn+1 โ UnโฉVn+1 .
Then Un โVn for all n and so { Un:nโN} is a base for U since
{Vn:nโN} is given to be a base.
NOTE 2.2
Having obtained a normalised countable base { Un:nโN}for U, we
can construct a pseudo-metric d on X as follows.
Set U0 = XรX. Note that Un โ Un-1 for all nโN.
12
Define d:XรX โ R in such a way that for each nโN, the set
๐ฅ, ๐ฆ โ ๐ ร ๐/๐(๐ฅ, ๐ฆ) < 2-n} will be very close to the set Un.Now we
construct a pseudo-metric d on X such that for each n, the set
๐ฅ, ๐ฆ โ ๐ ร ๐/๐(๐ฅ, ๐ฆ) < 2-n} will be between Un and Un-1.
We begin with a function ๐: ๐ฟ ร ๐ฟ โ ๐ defined by f(x,y) = 2-n
in case
there exists nโN. such that (x,y)โUn-1-Un.
If there exists no such n, it means (x,y)โ ๐โ๐=1 n and in that case we
set f(x,y)=0.
Here f (x,y)= f(y, x) โ x, yโ ๐ since all Unโs are symmetric. Also for
each nโN, ๐, ๐ โ ๐ฟ ร ๐ฟ/๐(๐, ๐) < ๐-n}=Un-1. Now for x,yโ ๐ define
d(x,y)=inf ๐๐๐=1 (xi,xi+1) where the infimum is taken over all possible
finite sequences {x0,x1,x2,โฆxn,xn+1} in X for which x0=x, and, xn+1=y.
Such a sequence will be called a chain from x to y with n nodes at
x1,x2,โฆxn.
The number ๐๐๐=1 (xi,xi+1) will be called the lengths of the chain .
Thus d(x,y) is the infimum of the lengths of all possible chains from x
to y.
LEMMA 2.3
The function d: ๐ ร ๐ โ R just defined is a pseudometric on the
set X.
Proof
Here f(x,y)=0 or 2-n
d(x,y) <2-n
13
โ d(x,y)โค f(x,y) for all x,yโ ๐
โd(x,y)โฅ0
If x=y, then f(x,y)=0
โd(x,y)=0
Now f(x,y)= f(y,x)
โd(x,y)= d(y,x)
Now only the triangle inequality remains to be established. Let x, y,
z โ ๐. Let S1, S2 and S3 be respectively the sets of all possible chains
from x to y, from y to z and from x to z. A chain s1โS1 and s2โS2
together determine an element of S3 by juxtaposition, which we
denote by s1+s2.
Let ๐ denote the length of the chain.
Now let J(S3) be the image of S1รS2 in S3 under the juxtaposition
function
+: S1 ร S2 โS3. Then we have,
d(x,y)+ d(y,z) = inf {๐(s1) : s1 โ S1} + inf {๐(s2) : s2โS2}
= inf {๐(s1) + ๐(s2) : (s1,s2)โS1รS2}
= inf {๐(s1+s2) : (s1,s2)โS1รS2}
= inf {๐(s3) : s3 โ J(S3) โS3}
โฅ inf {๐(s3): s3โS3}
= d(x,z)
Hence d(x,y)โฅ 0, d(x,y)=0 โ x = y
d(x,y)+ d(y,z) โฅ d(x, z)
14
hence d is a pseudo-metric on X
LEMMA 2.4
For any in teger kโฅ0 and x 0,x 1 ,x 2 ,โฆx kโX,
f(x 0,x k+1)โค2 ๐(๐๐=0 x i ,x i+1).
Proof:
We apply induction on k.
when k=0, the result is trivial.
Assume k>0, and that the result holds for all possible chains with less
than k nodes. Let x 0,x 1 ,x 2 ,โฆx k ,x k + 1 be a chain with k nodes. The
idea is to break this chain into smaller chains and then to apply the
induction hypothesis to each of them.
Let a be the length of this chain, that is, a= ๐(๐๐=0 xi,xi+1)
Here f(x i,x i+1) โฅ0 for all i
โ a โฅ 0 and a=0 only if f(x i,x i+1)=0 for every i=1,2,โฆ k
also f(x i,x i+1)โค a
now we show that f(x 0,x k+1) โค2a
now we make three cases
case1: a โฅ1/4
then 2a โฅ1/2, by the definition of f the largest value it can take
is ยฝ, since f(x,y)=2-n
, nโN
โด f(x 0,x k+1)โค1/2 โค2a
Case 2: Let a=0.
Then f(x i,x i+1)=0 โ i = 0,1,. . . , k.
We have to show that f(x0,x k+1)= 0
15
ie, to show that f(x0,x k+1)โ ๐โ๐=0 n
We decompose the chain x 0,x 1 ,x 2 ,โฆx k ,x k + 1 into three chains, say
x 0,x 1 ,x 2 ,โฆx r ; x r+1,...x s and x s+1,...x k,x k+1 where such that r and s
are any intiger such that 1โค r โค s โค k
note that each of these three chains has a lenth zero and less than k nodes.
so by induction, f(x0, xr) , f(xr,xs)and f(xs,xk+1) are all zero.
Hence for every nโN , (x0,xr) โUn , (xr,xs) โUn, and , (xs,xk+1) โUn
โ (x0,xk+1) โ UnโฆUnโฆUn
But UnโฆUnโฆUn < Un-1
There for (x0,xk+1) โ Un-1
So f(x0,xk+1)=0
Case 3: Let 0 < a < 1/4.
Let r be the largest integer such that f๐โ1๐=0 (xi,xi=1) โค a/2,if
f(x0,xi)> ๐/2, this definition fails and we set r =0 in this case. Then
0 โค r โค k
so each of the chains xo,โฆ,xr, and xr+1,โฆ, xk+1 has less than k nodes.
The first chain has length โค a/2.
Then the length of the second chain is also at most a/2 for otherwise
the chain xo,...,xr+1 will have length less than a/2 contradicting the
definition of r.
By induction hypothesis we now get, f(xo,xr) โคa, f(xr+1,xk+1) โคa
While f(xr, xr+1) โค a
Let m be the smallest integer such that 2-m โค a.
16
In particular, f(x0,xr+1) โค2-m
โ(x0, xr+1) โ Um-1
similarly (xr, xr+1) โ Um-1 and(xr+1, xk+1) โ Um-1
โ(x0, xk+1) โ Um-1โฆ Um-1โฆUm-1 โ Um-2
Hence f(x0, xk+1) โค 2-(m-1)
โค 2a
Hence the proof.
RESULT 2.5
A uniformity is a pseudoprime if and only if it has a countable base.
RESULT 2.6
A uniformity is metrisable if and only if it is Hausdorff and has
countable base.
PROPOSITION 2.7
Let U be uniformly generated by a family D of pseudometrices on a set
X. Then
i) each member of D is a uniformly continuous function from XรX to R
where R has the usual uniformity and XรX has the product uniformity
induced by U. Moreover U is the smallest uniformity on X which
makes each member of D uniformly continuous.
ii) Let Y be the powerset XD. For each dโD , let Xd be a copy of the set
X and let Vd be the uniformity on Xd induced by the pseudometric d.
let V be the product uniformity on
Y= Xdโ๐ท d. then the evaluation function f:XโY defined by f(x)(d)=x
for all dโD, xโX is a uniform embedding of (X,U) into (Y,V)
17
Proof
Proof of the first statement is clear
For ii) we let ฯd:YโX be the projection for dโD . Here for each
dโD
ฯdโฆf : X โXd is the identity function and is uniformly continuous because
the uniformity on X is U which is stronger than Vd. So by the general
properties of products f: (X,U) โ(Y,V) is uniformly continuous, clearly f is
one-one. Let Z be the range of f . We have to shoe that f uniformly
isomorphism when regarded as a function from (X,U) to (Z,V/Z). now we
prove that the image of every sub-basic entourage under fรf is an entourages
in V/Z. Take the sub-base s for U. A member of S is of the form Vd,r for
some r>0 and dโD. Then clearly (fรf) (Vd,r)is precisely Zโฉ(ฯdรฯd)-
1(Vd,r)which is an entourage in the relative uniformity on Z
PROPOSITION 2.8
Let (X,U) be a uniform space and D be the family of pseudometrices on
X which are uniformly continuous as function from XรX to R, the domin
being given the product uniformity induced by U on each factor. Then D
generates the uniformity on X.
proof
Let V be the uniformity generated by D By statement (i) of the last
proposition, V is the smallest uniformity on X rendering each dโD
uniformly continuous. So VโUโฆโฆ.(1)
Now suppose uโU. Let Ui be the symmetric member of U contained in U.
So U0=XรX . Define U2,U3......Un......... by induction so then
18
each Un is a symmetric member or U and UnโฆUnโฆUnโUn-1 for each nโN.
Then we get a pseudometric d: XรXโR such that for each nโN
Un+1 โ{(x,y)โXรX : d(x,y)<2-n
} โUn-1. Then d is uniformly continuous with
respect to the product uniformity on XรX. So dโD , then d generates the
uniformity U.
Put n=2, we get {(x,y) โ XรX: d(x,y)<1/4} โU1โU. Let S be the defining
sub-base for V. Then {(x,y) โ XรX: d(x,y)<1/4}โSโ V ,and so uโV , so
UโV โฆโฆโฆ(2)
From (1) and (2) we get U=V.
Thus D generates the uniformity U.
RESULT 2.9
Every uniform space is uniformly is uniformly isomorphic to a
subspace of a product of pseudometric spaces.
CORROLLARY 2.10
If (X,U) is a uniform space , then the corresponding topological space
(X,ฯu) is completely regular.
proof
from proposition 3.7 it follows that (X,ฯu) is embeddable into a product
of pseudometric spaces. Then (X,ฯu) is completely regular.
RESULT 2.11
A topological spaces is uniformisable if and only if it is completely
regular.
19
DEFINITION 2.12
The gage of uniform space (X,U) is the family of all pseudometrics on
the set X which are uniformly continuous as functions from XรX to R.
REMARK 2.13
In the view of the statement (i) in proposition 3.7 the uniformity is
completely characterized by its gage. Thus we have answered the two basic
questions through the result 3.6 and 3.11
21
DEFINITION 3.1
A net {Sn:nโD} in a set X is said to be a Cauchy net with respect to a
uniformity U on X if for every UโU there exist pโD such that for all m โฅ p
, n โฅ pin D (Sm,Sn)โD
PROPOSITION 3.2
A uniform space (X;U) is said to be complete if for every cauchy net in
X (with respect to U) converges to atleast one point in X (with respect to the
topology Iu)
PROPOSITION 3.3
Every convergent net is a Cauchy net. A Cauchy net is convergent if
and only if it has a cluster point.
Proof
Let {Sn:nโD} be a netin a uniform space (X;U) . suppose {Sn:nโD} is
converges to x in X. let UโU , then there exist a symmetric VโU such that
VโฆV โ U. Now V[x] is a neighbourhood of x in the uniform topology on X.
so threre exist a pโD such that for all nโฅp in D, Snโ V[x], ie, (x,Sn)โ V.
now for any m,n โฅp, (Sm,x) โ V and (x,Sn)โ V by symmetry of V
So (Sm,Sn)โ VโฆVโ U, since UโU was arbitrary , it follows that {Sn:nโD} is
a Cauchy net.
Now assume {Sn:nโD} is a Cauchy net. Let {Sn:nโD} converges
to x. Then x is a cluster point .
Conversely assume that x is a cluster point of a Cauchy net {Sn:nโD} in a
uniform space (X; U) . we have to show that Sn converges to x in the
uniform topology. Let G be a neighbourhood of x. Then there exist a
Uโ U such that U[x] โG. now we find
22
a symmetric VโU such that VโฆV โ U . then threre exist a pโ D such that
for all m,nโฅp in D,(Sm,Sn)โ V.Since x is a cluster point of {Sn:nโD} there
exist a qโฅp in D such that Sqโ V[x] . ie, (x,Sq) โ V. Then for all nโฅq we
have (Sq,Sn)โ V and (x,Sq)โ V .
So (x,Sn)โ VโฆV โ U.
Hence Snโ U[x] โG.
โ {Sn:nโD} converges to x.
COROLLARY 3.4
Every compact uniform space is complete
Proof
A compact uniform space means a uniform space whose associated
topological space is compact. We know that every net in a compact space
has a cluster point . Let (X;U) be a compact uniform space and {Sn:nโD} be
a cauchy net in it. Then (X;U) has a cluster point. So by the above
proposition, {Sn:nโD} is convergent. ie, every Cauchy net is convergent .
โ( X;U) is complete
โ every compact uniform space is complete.
DEFINITION 3.5
Let (X; U) , (Y,V) be uniform spaces and f : Xโ Y be uniform
continuous.Then for any Cauchy net S :Dโ X, the composite net fโฆS is a
Cauchy net in (Y,V).
23
PROPOSITION 3.6
Let (X,d) be a metric space and U the uniformity of X induced by d.
Then a net {Sn:nโD} in X is a Cauchy net with respect to uniformity U.
Moreover ,(X,d) is a complete metric space if and only if (X; U) is a
complete uniform space.
Proof
First suppose that {Sn:nโD} is a Cauchy net with respect to a metric d. Then
for every ํ>0, there exist n0โD such that for every m,nโ D, mโฅn0 and nโฅn0
d(Sn,Sm)<ํ.
Let Uโ U , then d(Sn,Sm)<ํ โ (Sn,Sm)โ U
ie for every ํ>0, there exist n0โ D such that for every m,nโ D, mโฅn0 and
nโฅn0 , (Sn,Sm)โ U
โ {Sn:nโD} is a Cauchy net with respect to the uniformity U
Conversely suppose that a net {Sn:nโD} is a Cauchy net with
respect to the uniformity U. let Uโ U then by the definition , for every UโU
there exist a pโ D such that for every mโฅp, nโฅ ๐ in D, (Sn,Sm)โ U
โ d(Sn,Sm)<ํ for all ํ>0
โ {Sn:nโD} is a Cauchy net with respect to the metric d.
Now we have to prove that (X; d) is a complete metrics space if and only if
(X; U) is a complete uniform space.
Since the metric topology on X induced by d is the same as the uniform
topology induced by U , convergens of a net with respect to the metric d is
same as that with respect to the uniformity U.
So to prove the above argument , it is enough to prove that every Cauchy net
in (X;d) is covergent if and only if every Cauchy sequence in (X;d) is
convergent. here a sequence is a special type of
24
net , so every net in (X;d) is convergent โ every Cauchy sequence in (X; d)
is convergent.
Conversely assume every Cauchy sequence in (X;d) is
convergent. Let {Sn:nโD} be a Cauchy net in (X;d). Now it is enough to
prove that {Sn:nโD} has a cluster point in X . Set ํ=2-k
, where k=1,2,โฆ. .
Then we obtain elements p1,p2,...pkโฆ. in D such that for each kโN , we
have,
(i) pk+1โฅpk in D and
(ii) for all m,nโฅpk in D , d(Sm,Sn)<2-k
Now consider the sequence {Spk} ;k=1,2โฆ. Since d(Spk,Spk+1)<2-k โ
kโN, and the series 2โ๐=1
-k is convergent., {Spk} ;k=1,2โฆ. is
Cauchy sequence in (X;d)
So {Spk} converges to a point say x of X. now we claim that x is the
cluster point of the net {Sn:nโD}. Let ํ>0 and mโD be given,
We have to find nโ D such that nโฅm and d(Sn,x)<ํ .first choose kโN so
that 2-k
< ํ/2 and d(Spk,x)< ํ/2
Now d(Sn,x) โค d(Sn,Spk)+ d(Spk,x)
< 2-k
+ ํ/2
< ํ/2 + ํ/2
< ํ
So x is a cluster point of the net {Sn:nโD}
โ every Cauchy net in (X;d) is convergent
โ every Cauchy net in ( X;U) is convergent
โ ( X;U) is complete uniform space.
25
DEFINITION 3.7
Given an induxed collection {( Xi,Ui):iโI} of uniform spaces . We
define a product uniformality on the cartesian product X= ๐๐โ๐ผ i
RESULT 3.8
Each projection ๐i : Xโ X i is uniformly continuous. Let S be the
family of all susets of XรX of the form ๐i-1
(Ui) for Uiโ Ui ,iโI , where ๐i:
Xร X โ X iร X i is the function ๐I ร ๐I defined by
( ๐I ร ๐I)(x,y) =( ๐i(x), ๐i(y))
PROPOSITION 3.9
Let ( X;U) be the uniform product of a family of nonempty uniform
spaces {(Xi,Ui):iโI}. Then a net S:Dโ X is a Cauchy net in (X;U) if and
only if for each iโI , the net ๐iโฆS is a Cauchy net in (Xi,Ui).
Proof
We know that the projection function ๐i is uniformly continuous where ๐i :
Xโ X i .
Now for any Cauchy net S:Dโ X , ๐iโฆS is a Cauchy net in (Xi,Ui) , since
๐iโฆS is the composite net.
Conversely assume that ๐iโฆS is a Cauchy net for each iโI.
Let S:Dโ X be a net, let S be the standard sub-base for U , consisting of all
subsets of the form ๐i-1
(Ui) for UiโUi and iโI. where ๐i: XรX โ Xiร Xi is
the function ๐i ร ๐I . Suppose U = ๐i-1
(Ui) for some UiโUi, iโI.
find pโ D so that for all m,n โฅp in D , (๐i(Sm), ๐i(Sn))โUi.
26
Such a p exist since ๐iโฆS is a Cauchy net
โ (๐I ร ๐I)(Sm,Sn)โ Ui.
โ ๐i(Sm,Sn) โ U Ui
โ (Sm,Sn) โ ๐i-1
(Ui)
โ (Sm,Sn) โ U โ m,nโฅp
So S is a Cauchy net in ( X;U)
REMARK 3.10
(X;U) is complete if and only if each (Xi,Ui) is complete.
DEFINITION 3.11
If (X;U) , (Y,V) are two uniform spaces, then the function f:XโY is
said to be an embedding, if it is one-one, uniformly continuous and a
uniform isomorphism, when regarded as a function from (X;U) onto
(f(x),V/f(x)).
THEOREM 3.12
Every uniform space is uniformly isomorphic to a dense subspace of
a complete uniform space.
Proof
Let (X;U) be a uniform space. We know that every uniform space is
uniformly isomorphic to a subspace of a product of pseudometric spaces.
Then there exist a family {(X i,di): iโI} of pseudometric spaces and a
uniform embedding,
27
f :Xโ ๐๐โ๐ผ i where the product ๐i is assigned the product uniformity,
each Xi being given the uniformity induced by di. for each iโI ,
let (X i*,di
*) be a complete pseudometric space containing (X i,di) up to an
isometry.
Then ๐i is a uniform subspace of ๐i* with the product uniformities, we
regard f as an embedding of (X;U) into ๐i* , which is complete.
Let Z be the closure of f(x) in ๐i*. then Z is complete with the relative
uniformity. Also f(x) is dense in Z.
Hence the proof.
DEFINITION 3.13
A uniform space (X;U) is said to be totally bounded or pre-compact if
for each UโU , there exist x1,x2,โฆxnโ X such that X = ๐๐๐=1 [xi].
Equivalently (X;U) is totally bounded if and only if for each UโU , there
exist a finit subset F of X such that U[F]= X.
THEOREM 3.14
A uniform space is compact if and only if it is complete and totally
bounded.
Proof
First assume that the uniform space (X;U) is compact. We have every
compact uniform space is complete. So (X;U) is complete.
Now compactness โ totally boundedness
โ (X;U) is totally bounded.
28
Conversely assume that (X;U) is bounded and totally bounded.we have to
prove that (X;U) is compact.
We know that a space is compact if and only if every universal net in it is
convergent. {Sn:nโD} is a Cauchy net in (X;U).
Let Uโ U, find the symmetric VโU such that VโฆVโ U.
By total boundedness of (X; U) , there exist x1,x2,โฆxkโ X such that X
= ๐๐๐=1 [xi].
Now for atleast one i=1,2,โฆk, {Sn:nโD} is eventually in V[xi] . for
otherwise the net will be eventually X - V k[xi] โ i=1,2,โฆk and hence
eventually (๐ โ ๐๐๐=1 [xi])=โ .
Thus for some i , {Sn:nโD} is eventually in V[xi]. ie, there exist pโD such
that โ nโฅp , Snโ V[xi] , ie, (xi,Sn)โ V .
So for all m,n โฅp in D, (Sm,Sn)โ VโฆVโ U. Thus {Sn:nโD} is a Cauchy net
in (X;U). Since (X;U) is complete {Sn} converges
โ (X;U) is compact.
RESULT 3.15
Every continuous function from a compact uniform space to a uniform
space is uniformly continuous.
PROPOSITION 3.16
Let (X;U) be a compact uniform space and V an open cover of X. then
there exist a UโU such that for each xโ X there exist a Vโ V such that
U[x] โ V
29
Proof
Given (X;U) is a compact uniform space and V an open cover of X.
Then for each xโ X , there exist UxโU such that Ux[x] is contained in some
members of V. Hence there exist a symmetric VxโU such that (Vxโฆ Vx)[x] is
contained in some members of V . The interiors of the sets Vx[x] for xโ X ,
cover X. So by compactness of X , there exist x1,x2,โฆxnโ X such that
X= ๐๐๐=1 i[xi] , where Vi denotes Vx i for i=1,2,โฆn.
Now let U = ๐๐๐=1 i , then UโU . Also let xโX , then xโVi[xi] for some i.
So U[x] โ Vi[xi]
โ Vi[Vi[xi]] , since xโVi[xi]
= (ViโฆVi)[xi]
โ some members of V
Since this holds for all xโ X, the result follows.
PROPOSITION 3.17
Let (X;U) be a compact uniform space. Then U is the only uniformity
on X which induces the topology ๐u on X.
Proof
If possible let V be any other uniformity on X such that ๐u=๐v
Let f : (X;U)โ (X;V) and g : (X;V)โ (X;U) be identity function. Then f and
g are continuous (in fact a homeomorphism) because ๐u=๐v .
30
Hence by result 4.15 , every continuous function from a compact uniform
space to a uniform space is uniformly continuous.
โ f and g are uniformly continuous
โ UโV and VโU
โ U=V
Hence the proof.
THEOREM 3.18
Let (X;U) be a compact uniform space. Let ๐u be the uniform topology
on X and given XรX the product topology. Then U consist precisely of all
the neighbourhoods of the diagonal โ๐ฅ in XรX
Proof
We know that every member of U is a neighbourhood of the diagonal โ๐ฅ in
XรX . Now it is enough to prove that every neighbourhood of โ๐ฅ is a
member of U. Let V be such a neighbourhood.
Without loss of generality, we may assume that V is open in XรX . Let B
be the family of those members of U which are closed in XรX. Then B is a
base for U, since each member of U is a closed symmetric member of U.
Let W=โฉ{ U:Uโ B}. We claim that W โ V .
For this suppose (x,y)โW, then yโ U[x] for all Uโ B .
Since B is a base , the family { U[x]:Uโ B} is a local base at x with respect
to ๐u.
In particular, since V[x] is an open neighbourhood of x , there exist a Uโ B
such that U[x] โ V[x]. Hence yโ U[x] โ V[x] โ yโV[x].
ie, (x,y)โ V. Thus W โ V.
31
since X is compact, so is XรX , sice V is open so XรX - V is closed.
B โช{ XรX - V } is a family of closed subsets of XรX and its intersection is
empty.
ie, there exist finitely many members U1,U2,โฆUn of B such that
U1โฉU2โฉโฆโฉUnโฉ(XรX-V)=โ
โ U1โฉU2โฉโฆโฉUn โ V
โ VโU.
Hence the proof.
32
BIBLIOGRAPHY
1. JOSHY.K.D โ Introduction to General Topology โ New age
International (p) Ltd.(1983)
2. AMSTRONG โBasic Topology SPR01 editionโ Springer (India)(p)
Ltd.(2005)
3. JAMES.I.M โIntroduction to Uniform Spacesโ Cambridge University
Press (1990)
4. JAN PACHL โUniform Spaces and Measuresโ Springer (2012)
5. MURDESHWAR.M.G โGeneral Topologyโ New age International (p)
Ltd. (2007)
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