LOCALLY MONOTONE MAPPINGS ON COMPACT METRIC SPACES

Kyungpook Mat.hemat.ical Journ혀

Volume ³², Nwnbσ 1, June, 1992

LOCALLY MONOTONE MAPPINGS ON COMPACT METRIC SPACES

Robert A. Johnson

1.

Introduction

In this paper we generalize the notion of a monotone mapping to pro- duce several related concepts. Our main concern is with developing a

usef비 definition of local monotonicity; our definition of a locally mono tone mapping

f

is based on the upper semicontinuous decomposition of the domain space o[

f

that is [ormed by the collection of the components of the p이 nt-inverses of

f.

We obtained our results

,

except for those in Section 3

,

unaware that a number of other definitions o[ local monotonic ity had already been given in the literature. Nevertheless

,

as Examples 1.1 and 3.2 indicate

,

the definition we propose should be more useful than the other definitions of local monotonicitYi generally

,

they are more restrictive than ours. Consider the following example.

Example 1.1. Let X

=

[0

,

lJ and let Y be the unit circle in the Euclidean plane. Define f : X • Y by f(x)

=

(cos2πx ， sin 2π) .

It will turn out that

,

by our definition (see Section 2)

, f

is locally monotone at each point of X

,

but under the definition given by T. Jvlack- owiak (see Section 3)

, f

is not locally monotone at the endpoints of X

All of our results concern functions defined on compa.ct metric spaces. In Section 2 we give some ba.sic definitions and establish several elementary results concerning locally monotone mappings. In Section 3 we compare our defìnition with other existing defìnitions of local monotonicity; we also compare the class of loca.lIy monotone mappings with some cl^a.sses o[

Received A pril 23, 1990

1

2 Robert A. Johnson

mappings common1y studied in connection with continua theory. 1n Sec- tion 4

,

characterizations of locally monotone and locally monotone

,

open mappings are given. 1n Section 5 a charaderization of locally monotone

,

qu때 open maps and various related resu1ts a.re given. We conclude with Sections 6 and 7; in Section 6 we prove that the composition of locally monotone ma.ppings defined on compact metric spaces is locally monotone

,

provided the given maps are quasi-open and satisfy a certain technical size condition

2. Pr e liminary Definit ions and Theorems

By a map

,

or mapping

,

f : X • Y we will always mean a continuous surjection. Unless otherwise specified

,

X

,

Y and Z will denote metric spaces

,

d will denote the metric on the appropriate space

,

and

f ,

9 and h will denote functions.

A co ηtinuum is a connected

,

compact metric space. A Peano contin- uum is a locally connected continuum. A map f : X • Y is monolone

([7]

,

p.26) if f-1 f(x) is a continuum for each x E X; f is called weakly monolone ([9], p.55i) if we require only that f-1f(x) be connected for each x E X.

From this point on we shall assume that all metric spaces are compact

,

although definitions will be given so as to apply to more general spaces We distinguish between monotone and weakly monotone functions because we will have occasion to restrict mappings to non-compact subspaces of the domain space

Let f: X • Y be a map. For x E X

,

1et C(x,j) denote the component of f-1 f(x) conlaining x. When no confusion can result

,

we write C(x) for C(x,j). We say f is locally monotone at x E X if (1) C(x) is compact and (2) there is an open set U containing C(x) such that U contains at most one component of f-1 f(z) for all z E X; the set U is called a pre-monolone neighborhood of x. A monotone neighborhood of x E X is any neighborhood V of x (not necessarily open) such that the restriction of f to V

,

namely f lv

,

is monotone. If f is locally monotone at each point of a subset A of X

,

we say f is locally monotone on A; if A = X

,

we say f is locally monotone. AIl other definitions of local monotonicity appear only in Sedion 3, and there they are clearly distinguished from our definition. Note that a monotone neighborhood of x must contain C(x)

,

while an open monotone neighborhood of x must be a neighborhood of C(x). A map f : X • Y is locallν one-to-one if given x E X there exists

Locally Monotone Mappings on Compact Metric Spaces 3

a neighborhood V of x such that

f

is one-to-one on V. The proofs of the following two theorems follow directly from the appropriate definitions.

The function in Example 1.1 is locally monotone at each point of [0,11 by Theorem 2.2

Theorem 2.1. lf f : X • Y is monotone

,

theπ f is locally monotone.

Theorem 2.2. lf f : X • Y is locally one-to-one

,

then f is locally monoto

’

^le.

For any subset A of a space X let ~(A) denote the number of compo- nents of A. A map

f :

X • Y is called n-monotone for some positive integer n if ~U-l J(x)) ::; n for all x E X and

"U-

¹ f(x))

=

ⁿ for some x. A map f : X • Y is open if the image under f of each open subset of X is open in Y. A map f : X • Y is called quasi-open at x E X ([101

,

p.9) if (1) C(x) is compact and (2) f(x) E intf(U) for all open sets U containing C(x). We say f is quasi-opeπ if f is quasi-open at each point of X. Let A" be a sequence of subsets of a space X. The limit

infeπ01" of An

,

denoted lim inf An

,

is the set of all x E X such that each neighborhood of x intersects all but a finite number of the sets An. The limit superior of An

’

denoted lim sup A"

,

is the set of all x E X such that each neighborhood of x intersects an infinite number of the sets An. The sequence An converges to a limit set A if lim inf An ⁼ lim sup An ⁼ A. In this case

,

we write lim A" = A or An • A.

Example 2.1. Let

X

be the unit circle in the complex plane

,

and define f :X • X by f(z)

=

^z2. Observe that f is open and locally monotone

,

but not monotone

Theorem 2.3. Let f : X • Y be a map and let x E X

A. lf f is locally monotone at x

,

then X contains an open monotone neighborhood of x

,

and conversely

B. lf f is locally monotone at x and quasi-open 011 X

,

then f(V) is open in Y J07. each open monotone neighborhood V of x

Proof (A) Suppose f is locally monotone at x; let U be a pre.monotone neighborhood of x. The required open monotone neighborhood of x is V = U{C(z)IC(z) C U for z E U}. Clearly

,

f is monotone on V. That V is open follows from noting that U is open and the fact that the collection

{C(z)lz E X} forms an upper sernicontinuous decomposition of X (see [1], Chapter 18). Conversely

,

suppose V is an open monotone neighborhood of x. Tl때 the compact set C(x) is a subset of V

,

and for each point

4 Robeπ A. Johnson

z E V we have that J-1J(Z)

n

V = C(Z). Thus

,

V is a pre-monotone neighborhood of x.

(B) Let V be an open monotone neighborhood of x. If z E V

,

then C(z)

c

V

,

since J is monotone on V

,

and J(z) E intJ(V)

,

since J is quasi-open on X. Thus J(V)

c

^{intJ(V) and J(V)} is open in Y

Theorem 2.4. lJ J : X • Y is locally monotoπ e， then J is n-monotone Jor some integer n

Proof For each x E X choose an open monotone neighborhood μ 。f

x. As X is compact

,

the collection of these open sets contains a finite subcover

Vi,'

^{‘ .}

,

Vk of X. Since

J

is monotone on V; for each i

,

i t folJows that n

<

k

If A is a subset of a metric space X and e

>

0

,

then N

,

(A) denotes the set of all points in X each of which Iies a distance less than e from some point of A. The proof of the next theorem uses the following well-known result.

Lemma 2.l. A mappi째 J :X • Y is open iJ and only iJlim J-1 J(xn)

=

J-1 J(x) Jor each point x E X and each sequence xn ^• x.

Theorem 2.5. lJ the opeπ mapping J : X • Y is not locally monotone at a point x E X

,

then there is a point t E C(x) such that J is not weakly monotone on any neighborhood oJt.

Proof Suppose J is open and not localJy monotone at the point x. Then for each positive integer n the neighborhood N1/n(C(x)) ^of C(x) contains at least two components of some point-inverse Pn = J-1 J(zn)

,

Zn E X. If C(x) C Pn for infinitely many n

,

then we may choose a sequence of points tn from the set J-1J(X) - C(x) such that t = 1imtn E C(x). Clearly

,

J is not weakly monotone on any neighborhood of t.

Otherwise

,

we may suppose that Pn

n

Pm q, whenever η ￥ m Without loss of generality (that is

,

by passing to a subsequence of Pn if necessary)

,

we may assume that the points Zn have been chosen so that Zn • Z E C(x). Then

,

by Lemrna 2.1

,

1imPn = limr1J(zn) = J-1J(X) For each n

,

choose a component Cn of Pn such that (1) Cn

c

^N1/n(C(x)) and (2) C

=

limCn

,

where C is a non-empty subcontinuum of C(x). For each n Jet Dn ⁼ Pn - C

’ ‘ ,

and let D ⁼ (lim Dn)

n

C(x). Then D is a non empty closed set

,

and C(x) ⁼ C U D. As C(X) is connected

,

C

n

D ￥ q,.

It now follows that

J

is not weakly monotone on any neighborhood of any point t E C

n

D

Locally Monotone Mappings on Compact Metric Spaces 5

3. Definitions of Local Monotonicity and Cornpari- son to Other Properties of Mappings

In this section only

,

our notion of local monotonicity will be denoted J /ocal monotoniαtν， or J-Im. Three other definitions of local monotonicity eXlst.

In 1970

,

P.T. Church ([2), p.162) gave the following definition. A map f: X • Y is locally monoto ηe at x E X if there exists an open set U containing x such that (1) flu is monotone and (2) f(U) is open in Y

In 1979

,

T. Mackowiak ([6]

,

p.18) gave the following definition. A map f: X • Y is locally monotone at x E X if there exists a closed neighborhood V of x such that (1) flv is monotone and (2) f(V) is a closed neighborhood of f(x)

In 1987

,

Z. Ditzian ([3], p.122) gave a definition restricted to real valued functions defined on the real numbers. A function

f :

R • R is

/ocallν monotonic if there exists an ^E

>

0 such that

,

on any interval of length ε ，

f

has at most one change of direction

、Io/e shall not discuss Ditzian’s deδnition. Church’s and Mackowiak’s definitions are considered only in this section

,

and will be denoted C-Iocal monotonicity

,

or C-Im

,

and M -Iocal moπotonicity， or M-Im

,

respectively.

The proof of the following theorem follows directly from the appropri- ate definitions.

Theorem 3.1. lf a map f : X • Y is C -1m at x E X

,

theη f is J -1m at x.

In general

,

no other implications exist among the three types of local monotonicity. The function given in Example 1.1 is J-Im at x 0 but it is not M-Im at x O. We see that this function is also C-Im at x

=

0 by considering any open set of the form U

=

[0

,

a) U (b

,

1)

,

where

o <

a

<

b

<

^1. One can easily check that the function 9 defined in the

next example is A1-lm at x

=

1

,

although it is not J-Im at x

=

^1.

Example 3.^1. Define 9 : [-2

,

2] • [0 ， 1]asf이lows

。/ι

<

-

‘떼

1

11

<

-

<

111 -

’J

κ --

1i

””

씨에

O

r-、--

시ν η 서 =

Now

,

^J'vf-Im • C-Im

,

together with C-Im • J-Im

,

would imply M- 1m • J-Im

,

which is not the case

,

as we have just seen. Thus

,

a map that is M-Im at a point need not be C-Im at that point. Our next example

6 Robeπ A. Johnson

is of a function which is J-Im at a point

,

but not C-lm at that point. A simple lemma wiU be used

Lemma 3.1. Let X = Y = [0

,

1]. D생ne f: X • Y as fo^/lows ifO::::

x:::: t

1f S s

^z

s 3 ^,

ν3

S

x

S

l.

Let V

=

(0

,

1) be a subsel of Y. Then there does not exist an open set R in X such that flR is moπotone and f(R)

=

V.

Proof Take V as given

,

and suppose R is an open f(R) = V. We will show that f is not monotone on

f -

¹

m

⁼

^{U ,}

^{힘 There are two cases to consider.}

set in

X

such that R. First, note that

Case L Suppose ~ E R. Then

,

since R is open

,

R must contain an interval of the form A =

G -

^E

^,

^~

+

^E) for some E

>

O. Clearly

, f

is not monotone on A

Case 2. Suppose ~ rf. R. Then ~ E R. Since R is open

,

R is equal to a disjoint union of open intervals. Let B be the open subinterval of R that contains the point ~. If ~ E B, then flB is not monotone. Suppose

~ rf. B. Let b = g.Lb.B. Then b 으

3

^‘ We treat the cases b

>

~ and b = ~ separately. If b

> },

then f-1 f(b) is a three-point set

,

{x，ν ， b}. Since b rf. R

,

we may assume that x E R. Then there exists a number ε

>

0 such that A =

( x -

^E

,

^X

+

E)

c

^{R. But then}

f l AU8

^is not monotone. If

b = ~， then ~ rf. R. Since

f-

¹

W

⁼

^{{i ,} n ^,

^{we have ~ E R. Then there}

exists a number é

>

0 such that A = (~ - ^E

,

~

+

^é) C R

,

but again

,

flAUB is not rnonotone.

1n either case

,

it foll。、'IS that flR is not rnonotone.

Example 3.2. Let X = [-2

,

1] and let Y = [-1

,

1].

by

-x -2

I

if -2 E z S -l;

if -1 :::: x :::: 0;

if

윤 ::::x:::: 감 +!갈，

Define

f :

X • Y

f(x) = ^2x 윤

3 2"

2x - 살T if

윤 + ^~2~

^{:::: x ::::}

^하 + ^{3 갈}

if 유

+

3 갈

::::x::::

삭î ， forn= 1

,

2

,

3

,.

We will show that

f

is J-lm at the point x = -2 but not C-Im at that point. Loosely speaking

,

we may describe

f

by saying

f

fixes [-1

,

^이

pointwise

,

it folds [-2

,

- 1] back onto [-1

,

이， and it zig-zags [0

,

1] onto

Locally Monotone Mappings on Compact Metric Spaces 7

[0

,

1] such that the number of zig-zags is infinite and

,

in Y

,

the sequence of zig-zags converges to 0 from the right

That

f

is J -1m at x

= -

2 follows by noting that [-2

,

-1) is a monoto아on ne밍ighborhood of x

Suppose U is an open set in X containing the point x = -2 such that f(U) is open in Y. Since 0 = f( -2)

,

there exists a number e

>

0 such that f(U끼f샤) contains

such that V

=

(α2-ⁿ-¹

서’

^{2-η) n C (-eι’껴셔g다). Let A}

⁼

^f-l(Vη); note that A

=

(2-n-l

,

2-n). It follows from the definition of f that f(X -A)nf(A) ⁼ <þ;

thus

,

the open set R

=

U

n

A is non-empty

,

and

,

furthermore

,

f(R)

=

V.

By Lemma 3.1

,

flR is not monotone. Since f is an at most 3-to-1 map

,

this implies that flu is not monotone

,

and hence f ^{is not} C-Im ^{at the} point x ⁼ - 2.

Next we consider the implications that exist among the three types of local monotonicity for maps which are quasi-open.

Theorem 3.2. If the quasi-open map f : X • Y is J-Im at x E X

,

then f is M-Im at x

Proof ^Suppose f is quasi-open and J-Im ^{at a point} x ^E X. ^Let U ^{be a} monotone neighborhood of x. Choose an open set 0 such that C(x) C

o

^{C 0 C U. As f} is quasi-open

,

we know that f(x) E intf(O)

,

which implies that f(O) is a closed neighborhood of f(x). Since the collection {C(z)lz ^E X} ^{forms an upper} semicontinuous decomposition of X ([1]

,

p. 470) we know that V ⁼ U{C(z)IC(z)

n

0 ￥ <þ} is closed set

,

and

o c

V

c

U. Thus, f(V) is a c10sed neighborhood of f(x) and flv is monotone

,

and so

f

^{is lvJ-lm at x.}

Theorems 3.1 and 3.2 together imply the next result.

Theorem 3.3. If the quasi-open map f : X • Y is C-Im at x E X

,

then

f

is M -1m at x

The function defined in Example 3.1 is quasi-open

,

so we have that a quasi-open M-Im map need not be J-Im. Furthermore

,

as in the unre- stricted case

,

M-Im • C-Im would contradict this

,

so a quasi-open map which is M-Im at a point need not be C-Im at that p이 nt. The quasi- open case is complete with the next theorem

,

which follows directly from Theorem 2.3

Theorem 3.4. If the quasi-open map f : X • Y is J-Im at x E X

,

then

f

is C -1m at x

8 Robeπ A. Johnson

Finally

,

we consider the implications that exist among the three types of local monotonicity for maps which are open. ln general

,

assuming that f is open and M -1m at a point does not guarantee that

1

will be J -1m at that point

,

as the next example shows. However

,

if

1 :

X • Y is open and M-Im on X

,

then it will be J-Im on X

,

as we prove next

Example 3.3. Define a subspace of the Euclidean plane as follows. For n

=

1

,

2

,

3

,"',

let An

=

{(x ， ~)1-1::::: x::::: -~，~::::: x::::: 1}j let A

=

UAnj let X be the closure of A in the plane. Define

1 :

X • I(X) by I(x

,

y)

=

(0

,

y). Let B

=

X - A. Clearly

, 1

is an open mapping. Observe that

1

is M-Im at all points of X except (0

,

0)

,

whereas

1

fails to be J-Im at every point of B.

Theorem 3.5. Jf the open map

1 :

X • Y is M -1m at each point of C(x) lor some point x E X

,

then

1

^{is J-Im at x} μe. ， on C(x)).

Proof Let

1

be openj we prove the contrapositive. Suppose f is not J-Im at x E X. Then

,

by Theorem 2.5

,

there is a point t E C(x) such that

1

is not weakly monotone on any neighborhood of t

,

and it follows that

1

is not monotone on any closed neighborhood of t.

We conclude this section with a short group of examples which shed some light on how J-Im maps differ from other commonly studied types of mappings. We need several defìnitions. See [6]

,

pp. 12-24

,

for further discussion.

A map f: X • Y is said to be

(i) an MO-map if there exists a space Z and mappings 9 : X • Z and h : Z • Y such that 9 is open

,

h is monotone

,

and

1 =

hgj

(ii) con.βuent if for each subcontinuum I< of Y each component of 1-1(I<) is mapped by 1 onto I<j

(iii) ψeaklν conβuent if for each subcontinuum I< of Y th"re exists a component C of 1-1(I<) such that I(C) = li끼

(iv) locally con.β따nt (Iocally weakly conβuent) if for each point y E Y there exists a closed neighborhood F of y such that Ilf-1(F) is a confluent mapping (weakly confluent mapping)

Among the maps already defìned

,

the implications indicated in Figure 3.1 exist. AIso

,

see [6]

,

p.28 and [4]

,

p.49. It is natural to ask how J-local monotonicity fìts into this picture. A monotone map must be J-Im

,

of course

,

but an open map need not be. The function of Example 3.4 is open

,

but it is not J-Im at x = 0

Locally Monotone Mappings on Compact Metric Spaces 9

monotone open

\、

/

MO ^{jν- 1m}

\ /

quas1-open

confluent

/ \

locally confluent weakly confluent

\ /

locally weakly confluent

Figure 3.1

Example 3.4. Defìne

1: [-1 , 1]

^• [0

,

1] by

I (x)

⁼ Ixl.

At the other extreme

,

a J-Im map need not be locally weakly confluent To see this

,

consider Example 1.1 again. 1n that example

, 1

wraps X

=

[0

,

1] once around SI

,

the unit circle

,

sending botb endpoints of X to y = (1

,

0) E SI. Let f{ be a closed subarc of SI that contains y in its interior. Then

1-

^{1 (f{)}

=

[0

,

이 U [b

,

1]

,

where 0 < a < b < 1. This choice of a subcontinuum ^f{ of SI shows that

1

is not confluent; moreover

,

it shows that

1

is not locally weakly confluent

,

since y = (1

,

0) E f{.

For the remainder of this section

,

we consider maps which are J-lm and quasi-open

,

which we denote J-Im-qo. They will play a major role in the rest of this paper as well

By Theorem 3.2

,

a J-lm-qo map must be M-Im. The converse is not true; in fact

,

the map of Example 3.1 is M-/m (hence quasi-open) and MO (see [6], p.16)

,

but it is not J-Im.

One may easily check that the map

1

of Example 3.5 is M-Im

,

but it is neither MO nor J-Im. To see that

1

is not an MO map

,

note that the map h defìned in Example 3.4 of [5]

,

p.104

,

is similar to

1,

and the

Robert A. Johnson

argument given there that h is not an MO map may be app1ied unaltered to show that

f

is not an MO map.

10

Example 3.5. Define

f :

[0

,

1] • [0

,

1] 잃 follows.

ifO ~ x ~~，

if ~

<

x

< L 1 :::: -:: ::::

~’

lf :

S

^I

S x

^,

if 우 <x <1 Z

A“₁

z - A*

1l qJ nU

/lllllι‘---，‘

ηn ““η =

Our final example is of a map defined using polar coordinates

Example 3.6. Let Y = 51 be the unit circle in the Euclidean plane. Let A = {(r，~)I1 ~ r ~ 2}. Let X = Y U A. Define a map

f :

X • Yas follows.

if

(r ,

0) E Y

,

if (r

,

O) E A.

One may check that f is J-lm and quasi-open (hence M-lm)

,

but it is not MO. To see that

f

is not an MO map

,

suppose there exists a space Z‘ an open map 9 : X • Z and a monotone map h : Z • Y such that

f =

^{hg. It} follows from [8], pp. 182-184

,

that Z is either an arc or a homeomorphic copy of X. If Z is an arc

,

it is well-known that there does not exist a monotone map h of an arc onto a circle. If Z = X

,

then

,

in fact

,

9 must be a homeomorphism

,

and we would not have the monotone map h in that case either

때

서

T --

( (

I /

l --

m

=

η r

n

Characterizations

In this section

,

we give characterizations of locally monotone maps and locally monotone open maps. A key characterization of locally monotone quasi-open maps appears in Section 5. The proof of Theorem 4.1

,

which extends Theorem 2.3

,

is routine

4.

Then the Y be a map and let x E X.

Theorem 4.1. Let

f

X following are equívalenl.

A. f ís locally monotone al x.

B. X coπ tains aπ open monotone neighbOl.hood of x.

C. There do not exíst sequences an and bn in X such that:

(1) liman

=

^a E C(x) and limbn

=

^{b E C(x)}

,

(2) f-^I f(an) ⁼ f-^I f(bn) for each n

,

^and

(3) C( an)

n

C( _bn) = <þ for each n -•

Locally Monotone Mappings on Compact Metric Spaces 11

D. There is a n umber ^Q

>

0 such that

,

for all positive é ::::: Q

,

마 (C(x))

intersecls at most one component of anν poínt-ínverse.

For a. function

f :

X • Y

,

any subset Q of X sa.tisfying the relation f-1f(Q) = Q is called an inverse set

Theorem 4.2. A map f : X • Y is locally monotone on X if and only if f is locally monotone on each inverse set Q in X.

Proof For the sufficiency

,

X itself is an inverse set.

For the necessity

,

let

Q

be an inverse set in X and let x E

Q.

Let V be an open monotone neighborhood of x. Let

y

E

vnQ.

Since

f

is monotone

。 n V a.nd Q is a.n inverse set

,

we have that f-1 f(y) n V n Q

=

^C(y

, J)

It follows that V n Q is an open monotone neighborhood of x in Q (see Theorem 2.3.A); that is

,

flQ is locally monotone at x.

It is of interest to note that

,

as the next example illustra.tes

,

a map

f

need not be locally monotone on X even though there exists an open cover H of X such that flG is weakly monotone for each G E H.

Example 4.1. We define a spa.ce X C [0

,

1] and a map f : X • f(X) as folJows. Let

A = H - ,

^1nln

=

¹

,

2

,

3 ... }; let

B

감， ~]; let

C =

H+ 숨In

=

1

,

2

,

3

, ... }.

Let

X =

^A U ^B

u

^C. For x E ^A

,

let~ f(x)

=

1 - ^x;

for x E B

,

let f(x)

=

~; for x E C

,

let f(x) x. Observe that f is wea.kly monotone when restricted to any element of the open cover H = {[O

,

~)nx， (~， l]nχ q ，~)} of X. However

,

if x E B

,

then C(x) = B

,

and

,

for sufficiently large n

,

every neighborhood of C(x) contains two components of f-1 f( ~ - 흙)， so f is not locally monotone at any point of B. Note also tha.t

f

is quasi-open

Theorem 4.3. Let f : X • Y be an open map and let x E X . Then the following are equivalent

A. f is locally monotone at x

B. There exists an open cover H ofC(x) such that flG is weakly mono- tone for each G E H .

C. There exists an open cover H of C(x) such that fl

c

is monotone for each G E H.

Proof

(A

• B). Let H = {V} for some open monotone neighborhood V of x.

(B• A). This follows directly from the contra.positive of Theorem 2.5.

(A• C). Let V be a monotone neighborhood of x. Since C(x) is com- pact

,

there is a number é

>

0 such that N<(C(x))

c

^V. Then 1ι (C(x))

12 Robert A. Johnson

is a premonotone neighborhood of x

,

and

,

as in the proof of Theorem 2.3.A

,

we may choose an open monotone neighborhood G of x such that C(x)

c

G

c

N

,(C(

x)). We will show t.hat f is monotone on G

Let z E 강 -G. Then

c(

z) C X -G. Let C

=

C(z)n(강 -G). As C is compact

,

we need only show that C is connected. Let ψ E C-{z}. Choose an integer m such that 1/m

<

d(ψ ， z)/2 ， N^1..(w) C V and N^1..(z) C V. For each integer n ~ m proceed as follows. Let On = f(Nl(z))nf(N l(ψ))

and let Y = f(z). As f is open

,

On is an open set which contains y. Since z is a limit point of G

,

we may choose a p。int zn E f-1(On)

n

Ni(z)

n

G Let Yn = f(zn). Since f

(J

-I(On)

n

Nl(ω)) = On and Yn E On

,

we may choose a point. ψn E f-I(Yn)

n

f-l(On)

n

Nl(W). Now

,

_Wn and Zn must lie in the same component of f-l (Yn)

,

because V contains both Wn ^and Zn and f is monotone on V. Let Cn

=

C(ωn)

=

C(zn)' Since Zn E G and f is monotone on G

,

it must be the case that Cn lies in G.

Let D lim sup Cn . We note that D is connected

,

{ψ ， z}

c

^D

c

강 - G

,

and D

c

^{C(z). Thus}

,

C contains D

,

which shows that C is connected.

(C • A). This follows directly from the contrapositive of Theorem 2.5

5. Local Monotonicity and Related Concepts

Recall the definition of n-monotonicit.y from Section 2. A map

f :

X • Y is called stπctIy n-monotone for some positive integer n if Ü

(J

^{- I} f(x))

=

n for all x E X. We say

f

is n-ê-monotone for some positive integer n and some ê

>

0 if ~(J -I (I<)) 으 n for all continua I< in Y of diameter ::; ê and

~(J-I( I<)) = n for at least one such continuum. We say

f

is strictly n-ê- monotone if

f

^is n-ê-monotone and ~(J-I( I<)) = n for all continua I< in Y of diameter ::; ê. We combine Whyburn

’

s notion of quasi-monotonicit.y ([8J

,

p.151) with the concept of confluency to obtain the next definition.

A map f: X • Y is said to be n -conβuent for some positive integer n if (1)

f

is confluent

,

(2) ~(J-I( I<)) ::; n for all contioua I< in Y and (3) there exists a ∞ntinuum I< in Y such that Ü

(J

-I(I<))

=

n. Given a map f:X • Y

,

for each positive integer i

,

let X

‘

.J

=

^{{x E XIÜ}

^(J

^-If(x))

=

^i}

When no confusion can result

,

we write X

‘

for X

‘

^.J Note that each X

’

^.J

is an inverse set.

The map

f

of Example 3.4 is open

,

2-monotone and 2-confluent

,

but it is neither locally monotone nor strictly 2-monotone. The results in this section show how these concepts are relatedj these relationships are partic

Locally Monotone Mappings on Compact Metric Spaces 13

ularly interesting in the cases when

f

is quasi-open or X is a continuum.

We wiII need the following standard lemma (see [5]

,

p.102)

Lemma 5.l. A map f: X • Y is quasi-open if and on/y if for each con- vergent sequence of points Xk • X in X it is the case that lim sup f-l f(Xkl intersects each component of f-1 f(x). Moreover

,

f is quasi-open if and on/y if for each convergent sequence of continua Kn • K in Y it is the case that lim s때 f-1(Kn) intersects each component of f-1(J().

Theorem 5.l. Let f : X • Y be a quasi-open map. The fol/owing aπ

equiva/ent.

A. f is /ocal/y monotone.

B. There exists a positive integer N such that X = X1 U ... U X ^N

X

^N _￥

</>,

and X

‘

is a closed (inverse) set f0^7" each i ’

Proof (A• B). Suppose f is locally monotone on X. From Theorem 2

.4, f

is n-monotone. Then N n

,

and X^N ￥ </>. Fix i

,

1 ::; i 으 N.

Let Xk be a sequence in X

’

such that Xk • X E X. As X is compact

,

we may

,

without loss of generality

,

assume that μ has been chosen so that D = limf-1f(xk)

c

^{f-1f(x) exists. If ~U-1f(x))}

<

i

,

then

,

since

~U-1f(Xk)) i for each k

,

an arbitrary neighborhood of some compo- nent of f-1 f(x) must intersect at least two components of f-1 f(Xk) [01

sufficiently large k. This contradicts the local monotonicity of

f .

Thus

,

"U- 1 f(x)) ~ i. Since D ⁼ limf-1 f(Xk) and ~U-l f(Xk)) = i for each k

,

it follows that ~(D) ::; i. By Lemma 5.1

,

D must intersect each component of f-1 f(x). Since D

c

^{f-1 f(x)}

,

it follows that ~U-1 f(x)) ::; "(D). Thus

,

x E X

’,

and hence X' is closed.

(B • A). Let x E

X.

Let

X

¹

, ... , X

^N be the pairwise disjoint

,

closed and compact inverse sets described in condition B. Then x E C(x) C X

’

[or some i

,

and there is a number é

>

0 such that N.(C(x))

n

X' = </>

for all j ￥ i. If i = 1

,

then N.(C(x)) is a pre-monotone neighborhood of x

,

so

f

is locally monotone at x. Suppose i

>

1

,

and suppose that

,

for each η，

Ni

(C(1)) intersects at least two components of some p。int­

inverse f-1 f(x n). Then

,

for n

>

~， f-1 f(x n) C X

‘.

As X is compact

,

we may again assume that limf-1 f(xn) C f-1 f(x) exists. It follows by a simple counting argument that lim f-1 f(x n)

n

C ⁼ </> for some component C of f-1 f(x) - C(x)

,

which contradicts the quasi-openness of f. Thus

,

for some integer n

,

Nι (C(x)) intersects at most one component of any point-inverse

,

so

f

^is locally monotone at x

Corollary 5.l. Let f : X • Y be a quasi-open /oca//y monotone map.

14 Robert A. John∞n

A. lf D is a coπnected subset of X

,

the>l there is an integer i such that DCX

’.

B. lf K is a connected subset of

Y ,

then there is an integer j such that K

c

^f(X').

Proo

f.

(A) This foJlows directly from Theorem 5.1.

(B) From Theorem 5.1

,

it follows that Y may be written as a urllon of pairwise disjoint c10sed sets: Y = f(X¹⁾ U ... U f(X^N)

,

and so any connected set K must lie within one of these sets.

Theorem 5.2. Let f X • Y be a quasi-open map. Then f is n- monotone if and only if f is n-conj대'uent.

Proo

f.

Suppose

f

is n-monotone. Quasi-open maps are confluent (see [4], p.49). But then the existence of a continuum J( in Y such that

~(J -1(K))

>

n would contradict the n-monotonicity of

f.

Conversely

,

the definition of n-conftuency implies n-monotonicity Theorem 5.3. Let the map f : X • Y be strictly n-monotone. Then f is locally monotone if and only if f is quasi-open.

Proo

f.

Suppose

f

is locaJly monotone. We wiJl show that the image under

f

of every open monotone neighborhood V in X is open in Y. Let V be an open monotone neighborhood in X. Let x E V and let y

=

^{f(x). We}

c1aim that y E intf(V). If this is not the case

,

then there must exist a sequence yk in Y - f(V) that converges to y, and, as X is compact, we may assume that yk has been chosen so that lim f-1 (Yk) = D ￥ 4> exists.

Then

limr1(Yk) = D

c

^{r 1f(x) =} C1 ^{U ... U} Cn

for the n components C\""

,

C_n of f-1f(x)

,

where C 1 = C(x). Since f-1(ydnv

=

4> for each k, it foJlows that DnC(x)

=

4>, since C(x)

c

^V

This is an immediate contradiction if n = 1

,

because in that case D

c

C(x). Suppose n

>

1. Since ~(J-1(Yk))

=

¹¹ for each k

,

it then foJlows by a pigeon-hole argument that there must exist a sequence of pairs of components

,

one pair from each set f-l(Yk)

,

∞nverging to a subset of a component C_j of f-1 f(x)

,

where 2 ~ j ~ n. But this contradicts the local monotonicity of f on C_j. Thus

,

f(V) C intf(V)

,

which implies that f is a ^quasνopen map.

Conversely

,

suppose that

f

is quasi-open. Since

f

is strictly n-monotone

,

it follows that Xn = X. The argument from the second half of the proof of Theorem 5.1 may be used here

,

unaltered

,

to show that

f

is locally monotone.

Locally Monotone Mappings on Compact Metric Spaces 15

We remark that the well-known result that a monotone map

J

is quasi- open follows from this theorem

,

since if

J

is monotone then it is locally monotone and strictly 1-monotone

Theorem 5.4. Lel J : X • Y be a quasi-open map. Then J ís slπctly η -monotone iJ and only iJ J is strictly n-e:-monolone Jor some e:

>

O.

Proof Suppose J is strictly n-monotone. By Theorem 5.3

,

J must be locally monotone

,

hence n-monotone

,

by Theorem 2

.4,

and n-confluent

,

by Theorem 5.2. Thus

,

J-'(K) has at most n components for all continua K in Y. Suppose

,

however

,

that for each e:

>

0

,

Y contains a continuum K. of diameter ~ e: such that ~(J-'(]iι))

<

n. Then we may choose a sequence of continua Ki in Y such that

(1) diam K

‘ s f

^{for each t}

,

(2) κ converges to a point y E

Y ,

(3) ~U- '(Ki)) = m

<

η for each i

,

and (4) A = limJ-'(Ki) C r l(y).

It follows that A has at most m components. Since J-'(y) has n

>

m components and

J

is quasi-open

,

this contradicts the second half ofLemma 5.l

The converse follows from the definitions.

Example 5.1. Let X ⁼ {x

,

y

,

z} be a three-point subset of any metric space Mj let Y

=

{y

,

z}. Define J : X • Y by J(x)

=

J(y)

=

y and J(z) = z. Then J is open

,

locally monotone and 2-monotone

,

but it is not strictly 2-monotone.

Theorem 5.5. Let J : X • Y be a quasi-open map and let X be a continuum. Then J is locally monolone ν aπd only iJ J is slrictly n- monolone.

Proof The sufficiency is from Theorem 5.3. For the necess띠， if J is locally monotone

,

then J is n-monotone. By Corollary 5.1

,

X ⁼

xn ^,

^{which says}

that J is strictly n-monotone

For the case when X is a continuum

,

Theorerns 5.3

,

5.4 and 5.5 may be combined to yield the following characterization.

Theorem 5.6. Let J : X • Y be a map and let X be a continuum. JJ J has any one oJ the first three oJ the Jollowing properties, then it has all oJ the remaining three or none oJ lhem.

A. J is /ocally monotone B. J is quasi-open.

16 Robeπ A. Johnson

C. f is strict/y n-monotone.

D. f is strict/y n-é:-moπotone.

We state a fìnal theorem

,

without proof

,

which is an easy consequence of our earlier theorems and the appropriate defìnitions (see

[1] ,

p

.4 8

and p.329). A function f : X • Y is called /ight if f-1 f(x) is totally discon- nected for each x E X.

Theorem 5.7. If f : X • Y is a light

,

/ocallν monotone map and X is a continuum

,

then the following are equivalent

A. f is open

B. f is stηctlν n-monotone.

C. f is a local homeomorphism.

D. f is a covering map.

6. Composition Theorem

In general

,

as the next example shows

,

it is not true that the compo sition of locally monotone maps is locally monotone

,

even if the maps are defined on Peano continua.

Example 6.1. Define an arc in the plane as follows.

X = {(s

,

t)ls = 0~1;0 ~ t ~ 1) U {(s

,

O)IO ~ 5 ~ 1).

Partition X into a collection G of subsets as follows. G will contain the element

{(O ,

0)

,

(1 ，이) and each two-point subset of X of the form {(0

,*),

(1

,*)}

for n = 1

,

2

,

3

,....

G will also contain all singletons {x}

in X which have not been used to form any of the two-point subsets fìrst mentioned. Let

f :

X • XjG be the quotient map associated with G.

Let Y = XjG. Partition Y into a collection H of subsets as follows. H will contain one element of the form z =

{J

((O

,

t))IO ~ t ~ 1}. H will also contain 때1 singletons {y} in Y which have not been used to form the element z of H. Let 9 : Y • Yj H be the quotient map associated with H.

Let Z

=

^{Yj H. Let h : X •} Z be the composition h

=

gf. Observe that

f

is locally monotone (J is locally one-to-one, apply Theorem 2끽， but not quasi-open

,

on X

,

while 9 is monotone on Y. However

,

h-¹h((0

,

0)) has an infinite number of components

,

so

,

by Theorem 2

.4,

h is not locally monotone on X

We now prove that the composition of locally monotone maps defìned on compact metric spaces is locally monotone

,

provided the given m^a.ps

Locally Monotone Mappings on Compact Metric Spaces 17

are quasi-open and one of the maps satisfies a certain size condition with respect to the other map. We wiU need several definitions and preliminary results.

Lemma 6.1. Let the map f X • Y be quasi-open and strictly m-

I';-monotone. If Dk • D is a sequence of continua in Y

,

each contin- uum having diameter ::::; 1';, then there is a subsequence Dk‘ of Dk such that lim f-l(Dk;) ￥ </> exists and has exactly m components

,

and such that each compoη ent of f- I (D) contains one and onlν one component of limf-l(Dd

Proof For each k

,

~(J-l(Dk)) = m. Since X is compact

,

we can choose a subsequence Dk, of Dk such that A

=

^limf-l(D

k.J =

^{AI U ... U Am ￥} </>, where the Aj are the m components of A. Let B

=

^f-I(D)

=

^{BI U .. .} UBm

,

where the Bj are the m components of B. Using Lemma 5.1 and the fact that A C B

,

we may index the components of B so that Aj C Bj for each

j

This lemma suggests a definition. A map

f :

X • Y is said to be a C -map, if there is a number 1';

>

0 such that for each convergent sequence of continua Dk • D in Y

,

each continuum having diameter ::::; 6

,

^{it is the}

case that there is a subsequence Dk

,

of Dk such that (1) limf-l(Dk

‘)

￥

&

exists

,

(2) ~(J-I(D))

= <<(J

-l(Dk.)) for each i

,

and (3) each component of f-l(D) contains one and only one ∞mponent of lim f- I (Dd. Clearly

,

the number 6 in the definition is not unique. If, however, a map

f

^satisfies

this definition for some number é

>

0 which we wish to make explicit., we shall say that f is a C -map at 1';

>

0

Lemma 6.1 then says that a quasi-open

,

strictly m-é-monotone map is a C-map at ε The lemma may be generalized to yield a theorem

Theorem 6.1. If f : X • Y is quasi-open and locally monotone

,

then f is a C-map.

Proof Consider the decompositions X X¹ U .. . U X’l,X ⁿ ￥ </>, and Y

=

f(X¹) U ... U f(Xη described by Theorem 5.1 and Corollary 5.1.

For any i such that X

‘

￥ </>, flx' is quasi-open and strictly i-monotone, so

,

by Theorem 5.4

,

flx' is strictly i-é;-monotone for some é;

>

O. Let éO = min{é

‘

IX

‘ -1

</>,1 ::::; i ::::; n}. Suppose Dk • D is a sequence of continua in Y

,

each continuum having diameter ::::; éo. It follows from Corollary 5.1 that D and all but finitely many of the Dk are contained in f(XI) for some j. As 1';0 ::::; éj and flxJ : XI • f(XI) is a quasi-open and strictly j-éj-monotone map

,

we have by Lemma 6.1 that flxJ is a C-map

18 Robert A. Johnson

at 1:

=

^1:0. It follows that

1

^{is a C-map at 1:}

=

^1:0

Let

1 : x

^{• Y} be a C-map. A map 9 : Y • Z is said to be 1:-

subordinate to

1

for some number 1:

>

0 ^if (1) for each point y E Y

,

^each

component of g-'g(ν) has diameter ::; 1:, and (2)

1

is a C-map at 1:

>

0 Suppose

1 : x

^{• Y is} quasi-open and locally monotone. Then

1

^{is a}

C-map for various values 1:

>

O. If we wish to use the 1:( = 60) constructed in the proof of Theorem 6.1

,

^{we will say that}

1

^{is a} C-map at the deηved valμe 1:; in that case

,

^{we will} also say that 9 : Y • Z is I: -subo여inate to

1

at the derived va/ue 1:.

Theorem 6.2. Let

1 : x

^{• Y} and 9 : Y • Z be quasi-open

,

locallν

monotone maps. JI 9 is I:-subordinate to

1

at the derived va/ue 1:

>

0

,

then h

=

gl is locallν monotone (and quasi-ope미.

Proof By Theorem 2.8 of [5J

,

we know that h is quasi-open. From (our) Theorem 2.4, there are integers n1 and n2 such that

1

is n

,

-monotone and 9 is nTmonotone. Since

1

is quasi-open

, 1

is nl-confluent (Theorem 5.2)^‘ Thus, h

=

gl is N-monotone for some N ::; n

,

ⁿ²

Consider the decomposition X = X 1,h U ... U XN,h described in The- orem 5.1. We will show that each X샤 is closed. Let Xk be a sequence chosen from Xi,h for some i such that Xk • X E X. As X is com pact

,

we may assume Xk has been chosen so that h-1 h(Xk) converges (to a subset of h- 1h(x)). Now

,

~(h-lh(Xk)) = i for each k implies that

~(lim h- 1 h(μ)) ::; i. Since h is quasi-open, it f，이lows from Lemma 5.1 that

~(limh-lh(x))::; i

Suppose ~(h-lh(x)) < i. Then there must exist a component J of h- 1h(x) and a sequence of pairs of distinct components Ak and Bk(Ak

n

Bk = 4> for all k) of h- 1 h(Xk) such that A = lim Ak

c

^{J and B = lim Bk}

c

J

Now

,

observe that E = I(J) is a component of g-lh(x)

,

and for each k I(Ak) and I(Bk) are cor매onents of g-1 h(xd. If I(A k)

n

_I(Bk)

=

^{4> for}

infinitely many k

,

this contradicts the local monotonicity of g. Otherwise

,

suppose Dk = I(Ak) ⁼ I(Bk) for all k. By Corollary 5.1

,

J

c

Xm^.J for some m ::; n1; we may assume Ak and Bk lie in Xm.J for all k as well.

Let P ⁼ IIX'셔 Xm.J • I(Xm.J); note that P is quasi-open and strictly m-I:-monotone. Let D = lim Dk' Each of the sets D

,

E and Dk (all k) has diameter ::; 1: since

1

is a C-map at 1: and 9 is I:-subordinate to

1

at the derived value 1:

>

O. Now

,

limp-1 (Dk) exists (since lim h- 1 h(Xk) exists)

,

and we have limp-1(Dk) C P-¹(D) C P-¹(E). Applying Lemma

Locally Monotone Mappings on Compact Metric Spaces 19

6.1 to the strictly m-e-monotone map F

,

we see that each of these sets has exactly m components

,

and each of the m components of F-¹(D) contains one and only one of the m corr때T

being distinct components 0아f liηimF-1(Dk) ， lie in distinct components of F-¹(D). However

,

A and B lie in the same component

,

namely J

,

of F- 1(E). Since F- 1(D) C F- 1(E)

,

this means that one corr때onen따t of

F-J(E터) contains two components of F-1(D)j but since D

c

E

,

F is confluent

,

and F- J (D) and F-J(E) each have exactly m components

,

this is irnpossible.

Thus

,

~(h-1h(x)) = i

,

x E Xi

,

h

,

and Xi

,

h is closed. By Theorem 5.1

,

h is locally monotone

,

and our proof is complete.

7. C o nclud ing

R emarks

We conclude wi th a final theorern and several questions

A space Y is oJ order ~ m at a point t E Y ([6], p.9) if for each e

>

0 there is an open set G containing t such that (1) diam G

<

e and (2) card bd G

<

m. The minimal cardinal number which satisfies this condition is called the 0여er oJ Y at t and is denoted ord

,

^Y. Our final theorem is a partial characterization of the locally monotone images of an arc

Theorem 7.1. Let J : [0

,

1] • Y be a Iocally monotone map. Then Jor each point t E Y

,

^{0예 Y ~ 2k}

,

^{where k = ~(J-1(t)).}

Proof Let t E Y. As J is n-monotone (Theorem 2.4), k = ~(J-1(t)) ~

n< ∞ Let C아1

,’

" '， C1α^{'k b야e} the k corr매one태n따t성s 0이f J-1끼(“띠t샤).Fori=I ，

... ,

k

,

let α = (ai

,

_bi) be a monotone neighborhood of C

‘

^j clearly

,

we can choose the end points ai and b

‘

^{of α such that (1) μ}

ⁿ

^{Uj = <þ for i}

^#

^{j and (2)}

f|u

, lS monotone.

Now

,

the monotone image of an arc is an arc ([6]

,

p.75). Thus

, {J

(Ui)ll ~ i ~ k} is a collection of k ar^C5 in Y such that t E J(Utl

n

.. nJ(Uk). Let T = [0

,

1]-(U1 ^{U ...} UUk). Clearly

,

ct = d(J(T)

,

t)

>

0 From this we see that for each positive e

<

ct there is an open set G in Y

。f diameter

<

é that contains t and for which bd G contains at most 2k points

We remark that in this theorem, if t is such that J-1 (t) conta.Îns either or both of the endpoints of [0

,

1]

,

then we can say ord

,

^{Y ~ 2k - 1 or}

ord

,

^{Y ~ 2k - 2}

^,

respectively^‘ It is clear from Theorem 7.1 that the locally monotone image of an arc must be a regular curve ([8], p.82).

We pose the following questions. What (more) can be said about