The open open topology for function spaces

THE OPEN-OPEN

TOPOLOGY

FOR

FUNCTION

SPACES

KATHRYN F. PORTER

Department

of Mathematical Sciences Saint

Mary’s

Collegeof California

Moraga,

CA.

94575

(Received February 20, 1991 and in revised form January 22, 1992)

ABSTRACT. Let

(X,T)

and

(Y,T’)

betopologicalspaces and let

F

C

yX.

Foreach

U

_

T,V

E

T’,

let

(U,V)

{f

E

F:

f(U)

C

V}.

Define theset

SOD

{(U,V):

U

Tand

V

T’}.

Then

SoD

isasubbasisforatopology,

Too

on

F,

which iscalled theopen-opentopology. Wecompare

Too

with othertopologiesand discuss itsproperties. Wealso show that

Too,

on

H(X),

thecollection of all self-homeomorphismson

X,

isequivalent to the topology inducedon

H(X)

bythe Pervin quasi-uniformityon X.

KEY WORDS AND PHRASES.

Compact-open

topology, admissibletopology,Galois space, Pervinquasi-uniformity, self-homeomorphism, quasi-uniform convergence.

1980AMS SUBJECTCLASSIFICATIONCODES. Primary 54C35,Secondary 54H13.

1. INTRODUCTION.

The type of set-set topology which will bediscussed here is one which can be defined as follows: Let

(X, T)

and

(Y, T*)

betopologicalspaces. Let UandVbe collections of subsetsof

X

and

Y,

respectively. Let

F

C

yX,

the collection of allfunctionsfrom

X

into Y. Define,for

U

U

and V

V, (U,V)= {f

E F:

f(U)

C

Y}.

Let

S(U,V)=

{(U,V):

U

U and

Y

e V}.

If

S(U, V)

is asubbasisfor atopology

T(U, V)

on

F

then

T(U,

V)

is calledaset-set

topology.

Oneof the original set-settopologiesisthe compact-opentopology,

TeD,

whichwasintroduced

in 1945 by

R.

Fox

[1].

Forthistopology, as onemaysurmisefrom the name, Uis the collection of all compact subsets of

X

and V

T*,

the collection of allopensubsets of

Y. Fox

and

Arens

[2]

developed andexamined the propertiesof this nowwell-known topology.

In

particular,it was shown that if

F

C

C(X, Y),

the collection of all continuous functions on

X

into

Y,

then

Tco

on

theconceptofadmissibletopologyfor functionspacesandwasinstrumental inthestudy of groups ofself-homeomorphisms andtopologicalgroups.

Other set-set topologies that have been ofinterest are: the

point-open,

topology, Tp,

also known as thetopology of pointwiseconvergence, in which Uis the collectionof allsingletons in

X

andV T*; theclosed-open topology, where

U

isthe collection ofallclosedsubsetsof

X

and the setV

T*;

and thebounded-open

topology

(Lambrinos

[3]),

where

U

is the collection ofall bounded subsets of

X

andagain, V T*.

In

section2 ofthis paper, weshallintroduce and discuss the open-open topology,

To,,,

for

function spaces. It willbeshown whichofthedesirableproperties

Too

possesses.

In

section3, the group of all self-homeomorphisms,

H(X),

endowed with

Too,

isdiscussed.

As

will be provenin section5,

Too,

on

H(X),

isactuallyequivalenttothePervintopologyof

quasi-uniform convergence

(Fletcher

[4]).

Oneoftheadvantages of the open-open topologyis the set-set notationwhich provides us withsimple notation and,

hence,

ourproofs are moreconcise than those using the cumbersomenotationof thequasi-uniformity. Pervinspaces will be discussed in section 4.

We

assumea basic

knowledge

of quasi-uniform spaces.

An

introduction to quasi-uniform

spacesmaybe foundinFletcher and Lindgren’s

[5]

orin Murdeshwar and Naimpally’s book

[6].

Throughoutthis paperweshallassume

(X,

T)

and

(Y, T*)

aretopologicalspaces.

2.

THE OPEN-OPEN TOPOLOGY.

Ifwelet

U

T

and V

T*,

thenSo,,

S(U, V)

is the subbasis forthe topology,

T,

on any

F

C

yx,

whichiscalled theopen-open

topology.

Wefirst examinesomeof the properties offunctionspaces the open-opentopologypossesses.

THEOREM 1. Let

F

C

C(X,

Y).

If

(Y, T*)

is

Ti

for 0, 1, 2,then

(F, Too)

is

Ti

for

=0,1,2.

PROOF.

Weshall show the case 2; the other cases aredone similarly.

Let

2.

Let f,g (5

F

such that

f #

g. Then there is some x (5

X

such that

f(x)

g(x).

IfY is

T2

thereexistsdisjointopensets O and Uin

Y

such that

f(x)

(5

U

and

g(x)

(50. Both

f

and gare

continuous, so thereareopen sets

V

and

W

in

X

with x (5

V

f’l

W, f(V)

C

U

and

g(W)

C

O.

f

(5

(V,

U),g

(5

(W, 0),

and

(V,

U)

t3

(W, O)

.

Thus,

(F,T,,o)

is

T2.

A

topology,

T’,

on

F

C

yx

iscalled anadmissible

(Arens

[2])

topologyfor

F

provided the evaluationmap,E:

(F,

T’)

x

(X,T)

-

(Y,

T*),

definedby

E(f,x)=

f(x),

is continuous.

THEOREM

2. If

F

C

C(X, Y)

then

Too

isadmissiblefor

F.

PROOF.

Let F

C

C(X,Y).

Let

O(5 T* andlet

(f,p)

(5

E-l(O).

Then

f(p)

(50. Since

f

is continuous, thereexistssome

U

(5

T

such thatp(5

U

and

f(U)

CO.

So,

(f,p)

(5

(U, O)

x

U.

Arens also has shown that if

T’

is admissible for

F

C

C(X,

Y),

then

T’

is finer than

Too.

FromthisfactandTheorem 2,itfollows that

Tco

C

Too.

3.

THE

OPEN-OPEN

TOPOLOGY

ON

H(X).

We

nowconsider

Too

on

H(X),

thecollectionof all self-homeomorphismson X.

Note

that

H(X)

withthe binary operation o, composition of functions, and identity elemente, is agroup.

Someof theset-settopologies previouslymentionedareequivalent undercertainhypotheses. For example,theclosed-open topologyisequaltothecompact-open topologywhenever

X

is com-pact

T,

the point-opentopologyisequivalent tothe compact-open topologyif allcompact subsets

of

X

arefinitesets. It isalwaysadvantageous toknow when topologiesare or are notequivalent.

In

particular,itiswell known that if

X

is

T1

then

Tp

C

To

andas wehave shown

To

C

Too.

When

are

To

and

Too

distinct? Onehypothesis under which these two topologiesare not equivalent is:

"Let

X

be

T

and Galois."

A

topological space is

Galois

provided that for each closed set, C C

X

and each point

p’E

X\C,

thereis an h

H(X)

such that

h(x)

xforallx

C

and

h(p)

#p.

Among

the spaces whichare

T2

Galoisarethetopologicalvectorspaces

and,

asFletcher

[7]

hasshown, locally

euclidean

T

spacesorhomogeneous0-dimensionalspaces whichhavenoisolatedpoints.

THEOREM

3. If

X

isa

T

Galois space then

Too

#

To

on

H(X).

PROOF. Let

X

be a

T

Galois space. Letx

X;

then

X

\ {x}

is open in

X

sothat

(X \ {x},X \

{x})

is open in

Too.

Butnotethat

(X

\

{x},X \

{x})

({x},

{x})

in

(H(X),To,,).

Let e be the identity map on

X

thene

e ({x},

{x}).

Claim:

({x},

{x})

Too

and hence

To

#

Too.

Let

(C,,

U,)

be a basic open set in

To

which contains e.

So,

C

C

U

for all

i----1

1,2,3, n.

Set

U0

X

and

Co

.

Let

P

{U}’=0

and

Q

{

C,

}’=0.

Define

Px

t3

{U

PIz

U

and

Qx

u{C

QIx

q

C}.

Let

L

Px

\

Qx. Note

thatz

L

and

L

is open in

X.

Case 1"

L

X: Then for each 1,2,3...,n,

U,

X,

so that

(C,,U,)

H(X)

and

i--1

f(c,,

u,)

(1},

{}1.

Case2:

L

#

X: Thus, since

X

isGalois, thereexistssome h

H(X)

such that

h(y)

y for all y

X\

L

and

h(x)

-

x.

So

h

({x},

{x})

and h

(L,L).

Letq

C

forsomej

{1,

2,

n}.

Ifq

L

then

h(q)

q C C

U

s. Ifq

L

thenx Cj fromwhich itfollows that

L

CUj. Thus,

h(L)

L

CU_s.

In

eithercase, h

(C,,U,)

and again

(C,,U,)

i’-I --1

Therefore,

({

x

},

{

x

})

To

and

To

Too.

Effros’ Theorem

(Effros

[8])

is awidely known and useful tool in thestudy ofhomogeneous

spacesand continuatheory. Ofitsseveralforms,the most popularis: If

X

is acompact

space

(H(X),T.),

and if

T

C T,on

H(X),

thenthe conclusion also holds on

(H(X),T).

Ancel

[9]

has asked thefollowingquestion: If the hypothesis of the Effros’ Theorem ischangedto

"X

is

acompact, homogeneous,Hausdorff space,’is theevaluation map on

(H(X),T,:o)

still open *? To

thisend,since

Too

C

Too,

wecould consider whetheraform ofEffros’Theorem would betruefor

Too

on

H(X).

Unfortunately,wediscoverthe following.

THEOREM4. Let

(X,T)

bea

TI

topological space. Then,for eachzE

X,

theevaluation map,

E:

(H(X),Too)

X

definedby

E:(h)

h(z),

isopenonlyif

T

is thediscretetopology.

PROOF. LetzEX. Then theset

O

((X\ {z}),(X

\

{z}))is

open in

(H(X),Too).

But

((X

\ {z}),(X

\

{z}))

({z},

{z}).

So.

E:(O)

{z}.

Thus

E

isopen for eachzfi

X

onlyif

X

is discrete.

Let

(G, o)

beagroup such that

(G,T)is

atopologicalspace,then

(G,T)is

atopologicalgroup provided thefollowingtwomapsarecontinuous.

(1)

m GxG Gdefinedby

re(g1, g2)

glog2

and(I) G Gdefinedby

(I)(9)

9

-1.

Ifonly thefirst map iscontinuous, then wecall

(G, T)

a

quasi-topological group

(Murdeshwar

and Naimpally

[6]).

It is not difficulttoshowthat if

(X,

T)

isatopological spaceand

G

isasubgroupof

H(X)

then

(G,

Too)

is aquasi-topological group.

However, (G, Too)

is not always a topological group as thefollowingexample

(Fletcher

[4])

shows: Let

X

Rand let thetopology on

X

be described as follows:

T=

{(a,b)

C R:a

<

0

< b}U{q,X}.

Let f,g:X

X

be defined

byf(z)

=-z and

9(z)

-.

Clearly,

f

and _{9 are} homeomorphisms on X. Note that

f(z)

f-’(z)

and

9-a(z)

-2z. Let

U

V

(-1,1).

Then

f

(U,

V) c=

Too.

Now define (I):

H(X)

--.

H(X)

by

(I)(h)

h

-1.

So,

f

(I)-I((U,

V)).

Claim: (I)is not continuous: Let O

(:’]((a,,

b,), (c,,d,))

bea basicopen set in

(H(X),Too)

which contains

f.

Thena,

<

0

<

b

and c,

<

0

<

d, for eachi. If z

_

(ai,bi)

then

f(z)

-z

_

(c,,d,).

So,

9(z)

-

.

(ci,

d,)

and, hence, 9 fi

O.

Thus,every basic open setcontaining

f

alsocontains 9. But

9-1(U)

V and so 9

(I)-’((U,

V)).

Therefore,

any basic open set containing

f

is not contained in

-’((U,

V)).

Thus, is not continuous and

(G,

Too)

isnot atopological group.

4.

PERVIN

SPACES.

A

topologicalspace,

(X,T),

is called a Pervinspace

(Fletcher

[4])

provided that for each finitecollection,,4, ofopen sets in

X,

thereexistssomeh

c=

H(X)

such thath

#

eand

h(U)

C

U

forall

U

E ,4.

Topologiesarerarely interestingifthey arethetrivialordiscretetopology. To thisend,we have:

THEOREM5.

(H(X), Too)is

not discrete ifand only if

(X, T)is

aPervinspace.

PROOF.

First, assume that

(X,T)

is aPervin space. Let

W

be a basic open set in

Too

which contains e; i.e. W

](O,,

U,)

where

O,

C

U,

for each 1,2, 3, n and

O,

and

U,

are open in X.

{O,

1,2,3,...,n}

isafinite collectionofopen sets in

X,

and

X

isaPervin space,

hence, thereexists some h (5

H(X)

such that h eand

h(O,)

C

O,

C

U,. So,

h (5 Wand h e. Since

(H(X), Too)

is aquasi-topologicalgroup,

(H(X), Too)

isnot adiscrete space.

Now assume that

(H(X),Too)

is not discrete. Let V be a finite collection ofopen sets in

X. Let O

f"]

(U,U).

Then, Oisa basic open set in

(H(X),Too)

which is not adiscretespace.

uEv

Hence,

thereexists h(50 with h

:p

e.

So,

(X, T)

is Pervin.

Fletcher

[4]

proved that thePervin topologyof quasi-uniform convergence on

H(X)

is not discrete if and only if

(X,T)

is Pervin.

In

order to prove this, Fletcher had to first introduce numerous definitions along with some mind bogglingnotation. The aboveproof, along with the few needed definitions involving

Too,

is anexampleof the simplification that thedefinition of

Too

offersoverthe quasi-uniformdefinition andnotation.

5.

THE

PERVIN TOPOLOGY OF

QUASI-UNIFORM

CONVERGENCE.

Recall that if

Q

isa quasi-uniformityon

X,

then the topology,

TQ,

on

X,

which has asits neighborhood base at x,

B

{U[x]:

U (5

Q},

iscalled the

topoloiy

inducedby

Q.

The ordered triple

(X, Q,

TQ)

iscalledaquasi-uniformspace.

A

topologicalspace,

(X,

T)

isquasi-uniformizable

provided there exists a quasi-uniformity,

Q,

such that

TQ

T.

In

1962, Pervin

[11]

proved that everytopologicalspace isquasi-uniformizable by giving thefollowingconstruction.

Let

(X,T)be

atopologicalspace. ForeachO(5

T,

definetheset

So

(OxO)t.J((X\O)xX).

Let S

{So

O

(5

T}.

Then S is a subbasis for a quasi-uniformity,

P,

for

X,

called the Pervin quasi-uniformityand,asiseasilyshown, Tp T.

If

(X,

Q)

isaquasi-uniformspacethen

Q

inducesatopologyon

H(X)

called thetopologyof quasi-uniform convergence

w.r.tQ,

asfollows: Foreach set U(5

Q,

letusdefine

W(U)

{(f,g)(5

H(X)

x

H(X):

(f(x),g(x))

(5 Uforallx (5

X}.

Then,

B(Q)

{W(U):

U (5

Q}

isabasisforQ*,

the quasi-uniformity of quasi-uniformconvergencew.r.t.

Q

(Naimpally

[12]).

Let

TO.

denote the

topologyon

H(X)

inducedbyQ*.

TQ.

iscalled thetopologyof quasi-uniform convergencew.r.t.

Q*.

If

P

isthePervinquasi-uniformityon

X

Te.

isthePervin topologyof quasi-uniform convergence.

At this timeone could, onceagain, prove that Tp. isnot discreteif andonly if

(X,

T)

is a Pervin space, this timeusing the quasi-uniform structure

[4].

Weleave thistothe reader.

We now show that the open-open topology is equivalent to the Pervin topology of quasi-uniformconvergence.

THEOREM

6. Let

(X,T)

be a topological space and let G be a subgroup of

H(X).

Then,

Too

Tp, onG.

PROOF. Let

(O, U)

be asubbasic open set in

Too

and let

f

(5

(O,

U).

Then

f(O)

C

U.

whichimplies thatg(x) U. Thus,g

(O,U)

and

W(Sv)

C

(O,U).

Therefore,

To

CTp..

Now

let V Tp. and let

f

V.

Then thereexists U 6- Psuch that

f

W(U)[f]

C V.

Since

U

P,

thereissomefinitecollection,

{U,

1,2,

n}

C

T

such that

5

Sv.

C

U.

Define

i----I

A

’(f-l(Ui)

Ui). A

isanopensetin

Too.

and

f

EA.

Assume

g E

A

andletx

for somej

{1,

2,

n},

then x

f-I(U,).

So, g(x)

6_.

Us,

hence,

(f(x),g(x))

If

f(x)

U

forsomej E

{1,2,...,n}

then

(f(x),g(x))

.

(X

\

U.i,X

C

5

Su,

C

U,

andit follows thatg

W(U)[f]

C_Vand

A

C

Y. So,

Too

T,..

ACKNOWLEDGEMENT.

The author would like to thank the Committee for Faculty and Curriculum DevelopmentatSaint

Mary’s

CollegeofCaliforniafor theirfinancial support.

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FOX, R.,

On Topologies forFunction

Spaces,

Bull.

Ame.r.

Math.

Soc,,

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(1945),

429-432.

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ARENS, R.,

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Groups, Amer.

_{J. Math.,68}

(1946),

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LAMBRINOS, P.,

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FLETCHER,

P.,

Homeomorphism

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LINDGREN, W.,

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