WORDS CONTINUOUS

ON THE CONTINUOUS FIXED POINT PROPERTY

F. CAMMAROTO DipartimcntodiMammatica UniversiuldiMessina 98166Sant’Agata 0Vlessina) ITALY

and T. NOIRI

Department^ofMathematics Yatsushiro College of Technology Yatsushiro,Kumamoto

866JAPAN

(Received July 7, 1988 and in revised form November 6, 1988)

ABSTRACT. In

this

paper,

wedefineand investigate the continuous retraction and^the-continuousfixed pointproperty.Theorem ofConnell

11]

and Theorem3.4 of

Arya

andDeb

[2]

are improved.

KEY WORDS ANDPHRASES.

5-continuous, 0-continuous,

weakly-continuous, semi-regular,

^almost- regular, Voted point property.

1980

MATHEMATICS SUBJECT CLASSIFICATION CODE.

54

C

10, 54

C

20.

0.

INTRODUCTIQN..

The notionof 0-continuous functions wasflu’stintroducedbyFomin i].Afterthat,thisnodonhas been widely investigatedintheliterature.

By utilizing

0-continuous functions,

Arya

^andDcb

[2]

definedand

investigated

the 0-continuous retraction,

the 0-continuous

fixed

point

property

and

the 0-continuous

homotopy.

On

theotherhand,in

[3]

and

[4]

thepresent authors haveindependentlyintroduced the notionof8--continuous functions.The

purpose

of this

paper

isto

apply

8-continuitytothe retraction and thefixedpointproperty.

In

Section2, we studythe retractionofatopological

space by

8-continuous functions. Section3 dealswith the fixedpoint propertyin relationto8-continuous functions. Themainresults of this

paper

are Theorems3.2and 3.3whichimproveTheorem of

I]

and Theorem3.4 of

[2],

respectively.

1.

PRELIMINARIES.

Throughoutthe present

paper, spaces

^willalwaysmean topological

spaces

on which noseparation axie:ns areassumedunless

explicitly

stated.

We

shall denotea

topological space by (X, x)

or simply by

X.

Let (X, x)

be a

space

and

A

asubsetofX.The closure of

A

andtheinteriorof

A

aredenoted by

,

and

’

^or

simply

,

and

k),

respectively.

A

subset

A

of

X

is saidtoberegular open (resp. regular closed)^if

A=(,)

(resp.

A A" ).

Thefamilyofregular opensetsof

X

will be denotedby

RO(X). A

point x of

X

is saidto be in the

S-

closure[5]of

A,

denoted by

CI(A),

^if

^{A V #}

^forevery

^{V RO(X)}

containingx. Asubset A is said to be 8-closed_[5] if

A=CIn(A).

^{Thecomplement of a-closed setis saidtobe 5-open.The}

topologyon

X

whichhas

RO(X)

as abasis is called the.semi-regularization of x and is denotedby

:’.

Itis obviousthat

every

element of

’"

^{is a}

⁵

^{open setof}

^(X, x). A space (X, x)

is saidtobesemi-regular if

x

:’. A space

(X,

x)

is said to be almost-regular

[6]

if for each regular closed set

F

^{and each} x

X F,

there exist

open

sets

U

and

V

such that

x U, F C V

and.

U r V

DEFINITION

1.1.

A

functionf"

(X, x)--- (Y, o)

issaidtobe -continuous 3, 4

(resp.

almost-continuous 7 ],O-continuous and

weakly

continuous

[8]

iffor each

x X

and each

open

neighborhood

V

of f(x), there exists an

open

neighborhood

U

of x such that f

U C V

(resp. f

U C V,

f

U C V

and f

(U) C V).

REMARK

1.1.

It

isshownin 2, 3, 9 thatthefollowing implications hold: 8-continuous almost- continuity

=

O-continuous =#weak-continuity, where noneof these implications^isreversible.

2. _5-

CONTINUOUS RETRACTIONS.

Arya

and Deb

[2]

defined a subset

A

ofaspace

X

tobe a 0-continuous retractof

X

if there exists a 0-continuous function f"

X

---)

A

such that f/A is the identity on

A. We

shall similarly define a 5-continuous retract.

DEFINITION

2.1.

A

subset

A

of

space X

is called a -continuousretract of

X

if there exists a 5-continuous function f"

X

---)

A

suchthat fistheidentityon

A,

that is,f(x =x forevery x A.

Andsuch a function f iscalled a 8-continuous retraction.

REMARK

2.1.

It

is obvious thatevery &continuousretract is a 0-continuousretract.However, every i-continuousretractisnotnecessarilyacontinuousretractasthefollowing exampleshows.

EXAMPLE

2.1.

LetX {a,

b,c,

d}

and

{,X, {a}, {a, b}, {a, c}, {a,

b,

c} }. Let A {a,

b,

c}

and f" (X, x

(A,

x /

A)

beafunction defined asfollows"

f(a)

a,

f(b)

b,

f(c)

candf(d) d. Then

A

is a -continuousretractof

X

butit isnotacontinuousretractof

X

since

f-I

a

})

x for a x /

A.

REMARK

2.2.

In Example

3.1 of

[2], Arya

and Deb showedthatevery 0-continuousretractsis not necessarily a continuous retract.

However,

this example is false.The 0-continuous functionf"^X---)

A

in

[2,Example 3.1]

is necessarilycontinuous since the subspace

A

is discrete and regular. Since every 5-continuous function is 0-continuous,

Example

2.1 also shows thatevery0-continuous retract is not a continuousretract.

We

shall investigate relationships between -continuousretractand continuousretract.

PROPOSITION

2.1. If

X

is a semi-regular

space

and

A

is a continuous retract of

X,

then

A

is a 5-continuousretractof

X.

PROOF.

Thisfollows from the fact thatacontinuousfunctionfromasemi-regular

space

is &continuous

[3, Prop. 1.5].

LEMMA

2.1. If

A

is either

open

or dense in a

space X

and

Ve

RO(X),then VC3A isregularopen inthe

subspace A.

PROOF.

If

A

isdense in

X,

then thisfollows from[10, p. 175, B)].

Next,

supposethat

A

isopenin

X

and

Ve RO(X).

Then, we have

V A(A)= _{( A} A)" (A)= V A A)"

V

A

A.

Moreover,

we have

V A

^c

A D (V A )’ A V A .On

the other hand,

*__

-VA AC (vX)’q ^{A VA WA.}

Th.erefore,

^{we obtain}

^Vr (A)=

Vrh

A

and hence

V A

isregular

open

in

A.

PROPOSITION

2.2.

Let X

be a semi-regular space and

A

eitheropenor dense in

X.

Then

A

is a continuousretractof

X

ifandonlyif

A

isa i-continuousretractof

X.

PROOF. From

Lemma 2.1, for

A

either

open

dense and

X

semiregular, x x* and

(’/A) (’*/A) C__ (’/A)* C__ (r/A).

Thcrcforc,

A

isscmircgularsothat f"

X A

is &continuous if andonlyif it is continuous.

PROPOSITION

2.3.

Lct X

be a

spacc

and

A

ascmi-rcgular

(rcsp.

almost-rcgular) subspacc of

X.

If

A

is a 5-continuous

(rcsp.

continuous)retractof

X,

thcn it is a continuous(rcsp. continuous)rctractof

X.

PROOF. Lct

f’X

A

be a 5-continuous rctraction and

A be.

scmi-rcgular.

Evcry

5-continuous function into ascmi-rcgular

spacc

is continuous

[3, Prop.

1.4 ]. Thcrcforc,

A

is a continuous rctract of

X.

Evcry

continuousfunction intoanalmostregular spaccis &continuous[3,

Prop.

1.8].Thcrcforc, thc sccond part follows.

THEOREM2.1. IfA is a -continuous retractofX andB is a -continuousrctractofA,thcn B is a &continuous retractof

X.

PROOF. Lct

f"

X.--> A

and

g A---B

^bc 5-continuous retractions. Thccompositcfunctiongof"

X--- ^B

is 5-continuous [3,

Prop. 3.2].

Moreover, we have gof)

(x)

g(f(x))

g(x

x for

every

xe

B C A.

Therefore, gof"

X.-o ^B

^{is a 5-}continuous retraction and hence

B

is a 5-continuous retractof

X.

THEOREM

2.2.

A

subset

A

ofa

space X

isa 5-continuousretractof

X

ifand onlyif forevery

space Y,

every 5-continuous function f"

A.-o ^Y

^{canbeextendedto a 5-}continuous of

X

into

Y.

PROOF.

Necessity.

Let

g:

X A

be a i- continuous retraction.Let

Y

beany spaceand f"

A---> Y

be any5-continuous function.Thencompositefunctionfog"

X---> Y

is i-continuous[3,

Prop.

3.2]. Moreover, wehave(fog) x f (g(x)) f(x for every xe A. Therefore,f ogis an extension of f.

Sufficiency. Let A

A

..-->

A

be the identity function on

A.

Then

iA

^is

-

continuous and henceby the hypothesis there exists a

&

continuous function g"

X--o A

such that

g/A

_A.Therefore,

A

is a

5-

continuous retract of

X.

THEOREM

2.3. If

A

isa

5-

continuousretractofaHausdorff

space X,

then

A

is 5-closed in

X.

PROOF. Let

f:

X A

be a

5-

continuous retraction.

Suppose

that

A

isnoti-closed in

X.

Thereexists apoint xe

C1, ^{(A) A.}

^Since

^x e ^{A, f(x} : x

and hencethere exist

open

sets

U

and

V

suchthat

x

e

U, f(x

e

V

and

U V

hence

U V .Let W

be

any

regular

open

set

containin.g

^x.Then

^{U W}

^is

a regular open set containing x.Sincexe

CI

(A),

U

W

A :

^q).Let a e

U W

A, then

f(a)

ae

U

and hence

f(a) e ^V.

Thisshows that

f(W) q V

for any regular openset

W

containing x.

This contradicts thefactthat f is 5-continuous.

3.

THE

5-CONTINUOI,/S

FIXED

PQINT pRQPERTY,

Arya

and Deb

[2]

defined a space

X

to have the 0-continuous fixed point property if,forevery 0-continuous function f"

X--

^X,there exists anxe

X

suchthatf(x x.Weshall define the 5-continuous (resp. weakly continuous)fixedpointproperty asfollows

DEFINITION

3.1.

A

space

X

is saidtohave the5-continuous(resp. weakly continuous fixed point property, briefly denotedby i

cFPP

(resp. wcFPP),ifforevery5-continuous(resp. weakly continuous) function f"

X---> X,

there existsan xe

X

such that f(x x.

REMARK

3.1.

It

is obvious that aspacewiththewcFPP hasnecessarily the 0-continuous fixed point property and a

space

withthe 0-continuous fixedpointproperty has both the

FPP

^{andthe fixed}point property.

We

giveanexamplethat aspace

,,,

lththefixedpoint property nednothave the 8cFPP.

EXAMPLE

3.1.Let X a,b,c and z

q,

X, a}, a, b}, a,c }.^]’henthespace (X, ,)hasthe fix pointproperty[2, Example 3.2].

Now,

let f:(X, ) + (X, z)^beafunctiondefinedby

f(a)

f(c)=bandf(b).

Then f is S-continuousbut doesnotafixedpoint. Therefore, (X,) doesnothave theFPP.

RERK

3.2.

We

needthefollowingtwo

spaces

which we wereunabletoobtain:

(1)

aspacewhichhas

FPP

but dsnothavethe fixedpointproperty.

(2)^aspacewhichhas the 0cFPP but doesnothavethewcFPP.

TOM

3.1.

t

A eitheropenordensein aspace X. "if

X

has the

FPP

and

A

isa &continuous rewactofX,then

A

has the

FPP.

PROOF. t

f:

A

₊

A

any S-continuous function. Since

A

is a 6-continuous rcmct of X, by Theorem 2.2 fcan extendedto a &continuous function F

X

^A.Let

A+ X

the inclusion. IfV isaregulopen^setof

X,

thenj;(V)

A

V isregular openinthe

sdbspace A

by

Lemma

2.1.Therefore,

F - ^{0 (V))= 0oF)(V)}

^is

^&open

ⁱⁿ

^X

and hencejoF:X

+X

^is &continuous. Since

X

has the

FPP,

x

0oF) (x)

=j

(F(x))

=j

(f(x))

f(x) for somexe

AC X.

Thisshowsthat

A

has the

FPP.

Thefollowing theorem is aslightmificationof Theorem of 11].

TOREM

3.2.

t

(X,) an almost-regular space with the 8cFPP. If is a topology for

X onger

than and

(*)= ()

tbr

eve ^G

^{e o, then}

^{X, o)}

has the fixpointproperty.

PROOF. Suppose

that f: (X, ) + (X, ) ^is any continuous function. t g (X, )+ (X, z)and h (X,g)+ (X, z) the functions defined by g(x h(x f(x

tr eve

^{xeX. Let (X, z)_)(X,)}

theidentityfunction.

en,

since C

,

is anopen

b0ection.

^{Moreoversince f ogis}continuous,g is

contuous. Next,we

shall show that h is &continuous.Let xe

X

andh(x )e

Ve

RO(X, ).Since(X,) isNmost-regul,there exists

G

e such that h(x)e

G C

⁽⁾

C

V.

Smcc

g iscontinuous,

ga

(G)e oand h

-

^(G)

^{P (G) g [G).}

^{Therefore, h}

^(G)

^f

^(G)

e o and hence,utilizing continuity of f we obtNn

{*)

_e

_{Ue RO}

_{X, andh}

_{C V}

Ths shows that h s 8continuous

Now, wesetU=h-(G)

,thenwehave

x )" (U)

^".

Since(X, ,) has

e FPP,

there existsxe

X

suchthatx h(x f(x ). Thisshowsthat (X, o)has the fix pointproperty.

COROLLARY

3.1 (Connell 11 ).

Suppose

(X,,) ^{is a}regularspacewiththe fixpointproperty. If o is atopologyforX,

zC

^gd

() G

^(*)for each

G

e

,

then(X, o)has the fixpointproperty.

PROOF. It

isshown in[3,

Corollau

^{1.8]flat if}

^Y

^{isregular,then f}

^{X Y}

^is 8-continuous if andonly if is continuous. Since

eveu

regular spaceisahnost regular,this is an immediateconsequence oftheorem3.2.

We

shlgivealemmawhich will used in theproofof

tte

finaltheorem.

LEMMA

3.1.

t

f

X

+

Y

and g

Y Z

be functions:

(1)

f iswetlycontinuousfandonlyif

f- (V) C f (V)

foreachopenset

V

of

Y.

(2)

If compositego f:

X Z

isweaklycontinuousandg

Z

is anopen bijection,^then fis

wey

continuous.

PROOF. Statement (1)

is

eorcm

7 of[

121. We

shall showStatement (2) by utilizing Statement (1).

Let

V any opensetof Y.Since g isopen,gCV)isopenin

Z

and

(go

0

^(g(V))

^C

^(go

-

^(g(V)). Since g s bjecnve, (g o

f) (g(V))

f (V). Moreover,since g isopen,

(g

o

0 (g(V))

f

(g (g (V))) C

f

gi- ’g(9)) f (V).Consequently,

we obtain ^";

(V) C

^q

CV)

and hence fiswe&lycontinuous.

e

following theoremisanimprovementof[2.Tlcorem3.4]and 11, Theorem

!.

THEOM

3.3.Let(X,z)bearegulspacewith the fixed point property. If isa topology for

X

stronger than z

and()

⁽tbr

eveu ^Ge , en

(X, o)has thewcFPP.

PROOF. t

f;(X, ) (X, o) be any weakly cotinuous function. Let g;(X, o)+ (X, ),

h:(X, ’r)

(X,

"r)

^and i: (X,

’r) (X, o)

be the same functions as in Proof ofTheorem 3.2. Since :f=i o

g

is weaklycontinuous and is an

open

bijection,

g

is

weakly

continuous

by Lemma

3.1.Since (X, :)isregular, giscontinuous

[8]. Next,

weshall showthat h iscontinuous.

Let x

e

X

and

V

be anopen setof (X,

")

containingh(x). Since

(X, x)

isregular, thereexists

G

^e a: suchthat h(x

G C ()C V.

Since g is continuous,

g-I (G)

o and

ha(G) fl(G) g

(G). Therefore,we haveh

(G) fl (G)

o.

Sind6"f

is

weakly

continuous,

by

Lemma 3.1

f -t) C f (0o (05). It

follows fromthe sameargument asⁱⁿ Proof ofTheorem3.2that h is continuous.Since (X, a:) ^hasthefixedpoint property,thereexists apoint x

X

suchthat x

h(x f(x ).

Thisshows that f hasthe fixedpoint property.

COROLLARY

3.2

(Arya

andDeb

[2]).

If

(X,

a:)is aregular spacewiththe fixedpointproperty and if o

((o)___

isatopologyfor

X

stronger than x such that

(

⁽⁾for each

G

e

,

then(X, a)hasthe 0-continuous fixedpointproperty.

ACKNOWLEDGEMENT

This

paper

waswrittenduringthe second authorstayedin MessinaUniversity for

May

and

June

1988.

He

wouldliketothanktoC.N.’tl.and Messina University for itshospitality.Thesecond author’sresearch wassupported by

M.P.I.

"fondi40%"

(ITALY).

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Math.

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