AND WORDS

Internat. J. Math. & Math. Sci.

VOL. 21 NO. 3 (1998) 459-462

459

ONCOUNTABLECONNECTEDHAUSDORFF SPACESINWHICH THE INTERSECTIONOFEVERY PAIROF

CONNECTED

SUBSETSISCONNECTED

V. TZANNES

Department of Mathematics University of Patras Patras 26110Greece

(Received July 2, 1996 and in revised form December 28, 1996)

ABSTRACT. We prove ^{that a}countable connected Hausdorff space ^{in which}the intersection of every pair of connected subsets is connected, cannot be locally connected, ^and also that every continuous functionfromacountable connected, locally connected Hausdorff space,^{to acountable} connected Hausdorffspacein whichtheintersectionofevery pairof connected subsetsisconnected, isconstant.

KEY WORDS AND PHRASES. Countableconnected, locally connected.

1992 AMS SUBJECT CLASSIFICATION CODE. 54G15, 54F55, 54D10, ^54D05.

1. INTRODUCTION.

TheproblemofexistenceofcountableconnectedHausdorff space inwhich theintersectionof every pair of connected subsets is connected was posed by ^{(vid in}

[1],

^{and was answered in}

[2].

Recently, Gruenhage

[3]

assuming the^continuumhypothesisconstructedaperfectly normalspace in which the only non-degenerate connected subsets of it, are the cofmite sets. Also assuming Martin’ s Axiomhe constructed a completely regular ^{and a}countable Hausdorffspace with this

property. Obviously,inthese spaces theintersection ofeverypairofconnected subsetsisconnected.

Noneofthespaces in

[2]

^and

[3]

^is locally connected,^orhas adispersion point.

Weprove that acountable connected Hausdorffspacein which the intersection ofevery pair of connected subsets is connected, cannot be locally connected, and also that every continuous function fromacountableconnected, locallyconnected Hausdorffspace, ^toacountable connected Hausdorffspace inwhich theintersection of every pair ^of connected subsetsis connected, iscon-

460 V. TZANNES

stant. Both these results hold in aHausdorff connected space ^{with a}dispersion point: The first is obvious and the second, fornot necessarily countable spaces, ^was provedby Coppin in

[4].

^Im- provements ^of Coppin’s result, ^as well ^as results concerning ^the constancy ^offunctions between twospaces,canbe foundin thepapers by Chew and Doyle

[5],

andbySanderson

[6].

Let X be a connected topological space. A point ^{is called a}cu__t point ^of X if.the space X

\ (t)

^{isnot connected. Thus, it isa cut point dX, then the subspace X}

\ {t)

^{is theunion}

of two mutually separated sets

A(t),B(t). (Two

^sets

A,B

^{are called}separated ^{if ACt and}

n

B

.)

Obviously, if

A(t), B(t)

^{areconnected,the separationis unique. Let x,y EX. A cut} point dX is saidto separate thepoints x,y ifthe above sets

A(t),B(t)

^{canbe chosen so that}

x

A(t)

and /

B(t).

The set of cut points dX separating ^the points x,/will^{be denoted by}

E(x, ).

^Theempty^{set and}thesingletons^{areconsideredto be}connected. All spaces ^areasumed to havemorethanonepoint.

2. RESULTS.

PROPOSITION 1. Let X be aHausdorff connected spa:e such that

E(a,b) t ,

^{for every}

a,b X. Then thereexists acontinuous non-constant real valuedfunction on X, separating the pointsaand b.

PROOF. The proof^{is reduced tothe} Urysohn’ ^sLemmainthefollowingmanner: Forevery point

E(a, b)

^{there exist two sets}

M(t), M(t)

^{such that a}

M.(),

^bG

Mb(t), M.(t) M(t)

^U

{t}, Mb(t) Mb(t)

^U

{t}

adX

\ {t} M.(t)

^U

Ms(t).

Hence the sets

+EE(o,b)

aeboth dosed disjoint cointaining the pointsa,brespectively,and not containing any cut point d Xseparating ^thepointsa, b. Consequently,forevery point doftheset d

pitive

^{dyadic rational} numberswe candefineanopen set

(M,(t))(d)

such that if d

<

r, then

M,(t)(d))

^C_

M,(t)(r). ^But

then thefunction

f(x) inf{d’x

⁶

(M,(t))(d)},

^ifx

^F,

^{andjr(x) 1,ifx 6}

F

is continuous separatingthe

Points

^{a, b.}

PROPOSITION 2. Does not exist acountableconnected, locally connected Hausdorff space in whichtheintersectiond everypairofconnectedsubsetsis connected.

PROOF. As it is proved in

[?,

Theorem

9.1]

^a connected locally connected space X is a Hausdorffspacein whichthe intersectiond every pair(indeed^every collection) of connected sets is connected, i and only if no two

Point

^{dX are conjugate. That is, E(,y) }, for every}

x, y X.Butthen, Proposition 1 impliesthat there exists^anon-constant continuous realvalued functiononX,which isimpossiblefor countableconnected spaces.

PROPOSITION3. LetXbeacountableconnected Hausdorff spaceinwhich the intersection of everypair^ofconnected subsetsis connected. Then

(1)

^{The subset DofXt} every pointof whichXis notlocallyconnected, is dense.

COUNTABLE CONNECTED HAUSDORFF SPACES 461

(2)

^{The subset L} at every point of which X is locally connected is totally disconnected or

empty.

PROOF

(I).

ByProposition 2,D

#

^{).Henceat everypointz GX}

\ ,

^{thespaceXislocally}

connected and therefore if

Uz

^{is an}open connectedneighbourhood^{ofzfor which}

Uz

^f

#

^),then

Uz

^{is alsoa}locallyconnected spacein which the intersection of everypairofconnected subsetsis

connected, ^{which is}impossible,byProposition 2.

(2).

^Obvious.

THEOREM.

Every

continuousfunction fromacountableconnected, locally connectedHaus- dorf space, to aconnected Hausdorff space in which the intersection of every pair ^ofconnected subsets isconnected,isconstant.

PROOF. Let

f

^{be a continuous} non-constant function from X to Y. Obviously ^the space Z

/(X)

^{is countable} connected Hausdorffin which the intersection ofevery pair of connected subsetsisconnected. Letz, be distinctpoints^ofXsuch that

f(z)

f(y)and let

U,(,), ^Uj,()

^be

disjoint openneighbourhoods

of/(m),/(),

respectively. ^Since

X

is locallyconnected there exists an open connectedneighbourhood U, of such that jr(U,) C_

UI(,). ^If/(U,) {]’(z)}

then we considerthe set 24

{a X:/(a) =/()}.

^{Sincethe setA}

\ ,

^isnotempty,^itfollows that there exist apoint ^a ]

\

^{andaconnected}open

neishbourhood

^U,of a, such that

/(U,)

^C_

and

1"(o) {/(z)}.

Therefore thecomponent

6’j,(,) of/()

ⁱⁿ

j,(,)

^{is not asingleton.}

Consider the component K

of/(t)

ⁱⁿ

\

Cj,(,). IfK

{f(t)}

then for the component M of Y in X

\ f-(C,(,))

^{it holds that}

/(M’) {]’(t)}

^and

.f() {f(t)}.

^{Since the} subspace X

\/-(C,,))

^{islocallyconnecteditfollowsthat M’is}

open-and-closed (in

^X

\ f-(C/{,))),

^d

hence f

.f-(C(,)) #

^whichis impossible. ^Thereforethe component Kof in Z

\ 6’j,(,)

^is

not asingleton.

Thus, by

[8,

Vol.

II,

Ch.

V,

Theorem 5,

HI],

^fortheconnected subsets

Cj,()

^and

K

itfollows that the set

\/f

^isconnected and henceeither

(1) ( \/f) t,

^or

(2) (\K)f’I

o

(z)(\c)#.

Incase

(1),

^letp,/

( \/(’)

^Iq

K,

and/9

#

^{/.Thenfor theconnectedsubsets}

(Z \

andKitholds that

((Z \K)U

{p,/})^fl/f {p,/}^whichisimpossible^becausebyassumptionthe intersectionofeverypair ofconnected subsets ofZmustbeconnected. Therefore

(Z \ If)fl

^Kisa singleton. Weset

( \ K)fK {p}.

The setKis closed because ifaisalimit

Point

^ofKanda

g

K thenforthe connectedsubsets/ft.J

{a}

^and

( \ arc)

^thesubset

( \/f) (/f

^U

{a})

{a,p} must be connected,which isimpossible. Hence ifweconsider thecomponent^jklrof tⁱⁿX

\/-(Cj,(,))

then

f()

^{C/whichisalso}impossible because

Iq/-(Cj,(,))

Incase

(2)

^itcanbeprovedinthesame mannerasin case

(1)

^that

( \ K)fl

^isasingleton

andthat

Z\K

^isclosed. Weset

(Z\K)]- {g}.

Since

= ^KU{g)

^{itlollowsthat}

^(Z\K)\

is openwhichimplies that qis acut point ofthe space Z. Since q6 Z

\

Kit follows that ether

462 V. TZANNES

q

f(z)

^orq

:/: f(z).

^Ifq

f(z)

^{we consider}again thecomponent M^of /in X

\ f-*(Cf(,)),

^and

let a 6-

n f-*(Cf(,)).

^Then

f(M)

^{C_K, the point}

f(a)

^{is alimit} pointofK and

f(a)

That is

f(a)

q. But then there exists an open connected neighbourhood

U

^{of a such that}

f(U,,)

C_

U,(z)

^which implies that

f(U,,)

^C_

C.q,.).

^{Hence U,, C_}

f-(C.(,.))

^{which is impossible}

because

Ua

^r3M

.

^{If q}

^f(z)

^{thenobviously q6_}

E(f(z), f()).

Finally,observing^thatcase

(3)

^isreduced tocase

(1)

^or

(2)

^{weconcludethat}

E(f(z), f())

which is impossibleby Proposition 1.

REFERENCES

1.

(vid,

S.F. Acountable stronglyunicoherent space

(Russian),

^{Mat. Zametki 24}

(1978),

^no2, 289-294, ^303.

(Engl.

translation: Math.

Notes

24

(1978)

^no. 1-2,

655-657).

2.

Tzannes,

V. Three countable connected spaces, Colloq. Math.2

(1988)

^267-279.

3. Oruenhage, ^G. Spacesinwhichthenon-degenerateconnectedsubsetsarethe cofmitesets

(to

appear).

4. Coppin, C.A.Continuous functionsfromaconnectedlocallyconnected spaceintoaconnected spacewithadispersion point, Proc. Amer. Math. Soc.

32(2), (1972),

^625-626.

g.

_{ChewG.FandDoyle,P.H.On thepotencyofspaceswith}generalizeddispersion points under maps ^andfunctions, ActaMath. Acad. Sci.

Hungar.,

²⁹

(1977),

^51-53.

6. Sanderson, D.E.Criteriafor constancyoffunctions withalmosttotallydisconnected range^or domain, ActaMath. Acad.Sci.

Hungar.

35

(1980),

^33-36.

7. Whybn, G.T. Cut points in general topological spaces, Proc. Nat. Acad. Sci. 61

(1968)

380-387.

8. Kuratowski, K. Topology,Academic

Press,

1968.