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교육학석사학위청구논문

Peano 공간과 Hahn-Mazurkiewicz 정리

Peano Spaces and Hahn-Mazurkiewicz Theorem

2012년 2월

인하대학교 교육대학원

수학교육전공

이 슬 기

교육학석사학위청구논문

Peano 공간과 Hahn-Mazurkiewicz 정리

Peano Spaces and Hahn-Mazurkiewicz Theorem

2012년 2월

지도교수 송 용 진

이 논문을 석사학위 논문으로 제출함.

주심 김 재 문 (인)

부심 송 용 진 (인)

부심 정 해 원 (인)

본 논문을 이슬기의 석사학위 논문으로 인준함.

2012년 2월 일

i

국문초록

본 논문에서는 Peano 공간과 Hahn-Mazurkiewicz정리에 대해 연구하였

다. 본 연구에 필요한 기초적인 정의와 정리를 알아보고, 본 논문의 내용을

설명하는 데 필요한 주요한 성질들을 설명하였다. Hahn-Mazurkiewicz정리

는 Peano 공간이 될 필요충분조건이 그 공간이 단위 구간 의 연속 상(像)

이라는 것이고, 이 정리를 Cantor 집합을 이용해서 증명하였다.

ii

Abstract

We have studied Peano space and Hahn-Mazurkiewicz theorem. We

introduced basic definitions, theorems and properties needed for this

study.

Hahn-Mazurkiewicz Theorem states that a space is a Peano space if

and only if it is a continuous image of the unit interval . We have

proved the theorem by using the Cantor set.

1

Contents

국문초록................................................................... i

Abstract.................................................................. ii

1. Introduction........................................................ 2

2. Basic definitions and theorems....................... 4

3. Peano spaces and Hahn-Mazurkiewicz

Theorem............................................................. 11

References............................................................. 16

2

1. Introduction

A compact, connected, and locally connected metric space is called a

Peano space. Peano space has some history. About a century ago, when

mathematicians were first formulating, with a careful rigor, the concept

of "curve," Peano found a pair of continuous functions and whose

graph is 2-dimensional, filling up the square and its interior. This

example, surprising and almost paradoxical at the time, is commemorated

in the term Peano space. By the term "curve," mathematicians wanted to

mean a topological space which is "1-dimensional." For some technical

reasons they also wanted to define a curve as a continuous image of an

interval into a space such as ℝ . However, the space-filling curve inthe 2-dimensional plane constructed by Peano is a critical

counter-example to this idea, because the image of it is "2-dimensional."

This curve is called Peano curve.

The Hahn-Mazurkiewicz Theorem states that a Hausdorff space is

a Peano space if and only if it is the image of the unit interval

. By this theorem we are able to realize that a continuous

image of an interval cannot be called a curve,; it should be called a

Peano space.

It is obvious that a continuous image of the unit interval is a Peano

space. The converse part of the Hahn-Mazurkiewicz Theorem is not

easy to prove. The following theorem plays a key role in the proof of

this:

Theorem 2.18 Every compact metric space is a continuous image

3

of the Cantor set.

For a continuous map from the Cantor set to , we extend this map

to the whole unit interval by using the following theorem:

Theorem 3.4 A Peano space is uniformly locally path connected;

for each , there is a such that whenever , then

x and y are joined by a path of diameter less than .

In Chapter 2, we introduce basic definitions, theorems and properties

needed for the main result of this thesis. Theorem 2.18 and Theorem

2.19 are especially important.

In Chapter 3, we deal with Peano spaces and Hahn-Mazurkiewicz. We

first introduce three theorems related to Peano spaces. These theorems

are directed specifically to the proof of the Hahn-Mazurkiewicz theorem.

As mentioned above, extending a continuous map the Cantor set to a

Peano space is an essential part of the proof of this theorem.

4

2. Basic definitions and theorems

In this chapter, we introduce definitions and theorems which are

needed to explain Peano space and Hahn-Mazurkiewicz Theorem. They

will be helpful to understand major contents in chapter 3.

Definition 2.1 A topological space is called a Hausdorff space if

for each pair , of distinct points of , there exist neighborhood ,

and of and , respectively, that are disjoint.

Definition 2.2 Let and be topological spaces. A function

→ is said to be continuous if for each open subset of ,

the set is an open subset of .

Recall that is the set of all points of for which

; it is empty if does not intersect the image set of .

Definition 2.3 If and are topological spaces, a function from

to is a homeomorphism if and only if is one-one, onto and

continuous and is also continuous. In this case, we say and

are homeomorphic.

If is everything but onto, we call it an embedding of into ,

and say that is embedded in by . Thus, is embedded in

by if and only if is a homeomorphism between and some

subspace of .

5

Definition 2.4 If is a topological space, is said to be

metrizable if there exists a metric on the set that induces the

topology of . A metric space is a metrizable space together with a

specific metric that gives the topology of .

Definition 2.5 Let be a topological space. A separation of is a

pair of disjoint nonempty open subsets of whose union is .

The space is said to be connected if there does not exist a

separation of .

Definition 2.6 A space is said to be path connected if any two

distinct points can be joined by a path, that is a path which is a

homeomorphism between the unit interval and its image .

Definition 2.7 A space is said to be locally connected at x if for

every neighborhood of x, there is a connected neighborhood of x

contained in . If is locally connected at each of its points, it is said

simply to be locally connected . Similarly, a space is said to be

locally path connected at x if for every neighborhood of x, there is a

path connected neighborhood of x contained in . If is locally path

connected at each of its points, then it is said to be locally path

connected.

Definition 2.8 A simple chain connecting two points and of a

space is a sequence of open sets of such that

6

only, only, and ∩ ≠ ∅ if and only if ≤

Theorem 2.9 If is connected and is any open cover of , then

any two points a and b of can be connected by a simple chain

consisting of elements of .

Let be the set of all points of which are connected to

by a simple chain of elements of . Then is obviously an open set

and, since ∈ , is nonempty. We can prove the theorem by

showing is closed.

Let ∈ . Then ∈ for some ∈ and, since is open,

∩ contains some point . Now is connected to by a simple

chain of elements of . If ∈ for some , then the

smallest such produces a simple chain from to . If

∉ for any , pick the smallest such that ∩ ≠ ∅(e.g, is

such an ). Then , is a simple chain from to . Either

way, ∈ . ■

Definition 2.10 A continuum is a compact, connected Hausdorff

space. Among the continua we find many familiar spaces. Thus the

unit interval I , the circle , the torus (and, in fact, any

product of continua) are all continua. Our main goal is to find

topological criteria which will enable us to characterize the unit interval

and the unit circle as continua.

7

Theorem 2.11 If is a metric continuum with exactly two noncut

points, then is homeomorphic the unit interval I .

Proof. Let be a countable dense subset of not containing the

noncut points and . Note that :

a) has no smallest or largest element,

b) given and in with there is an element of with

.

We know that every countable totally ordered set with these

properties is order isomorphic, and thus homeomorphic, to the dyadic

rationals in the interval (0, 1). Let be an order isomorphism of

onto .

But each point of other than or is a cut point, dividing

into sets and with (i.e., whenever and

∈ ). It follows that ∩ and ∩ form a Dedekind cut

of the dyadic rationals, and thus uniquely determine an element of

(0, 1). Defining and , we have completed the job of

extending to what is obviously an order isomorphism, and thus a

homeomorphism, of onto . ■

Definition 2.12 A space is said to be compact if every open

covering of contains a finite subcollection that also covers .

Theorem 2.13 The image of a compact space under a continuous

8

map is compact.

Let → be continuous; let be compact. Let be a

covering of the set by sets open in . The collection

{ }

is a collection of sets covering ; these sets are open in because

is continuous. Hence finitely many of them, say

cover . Then the sets cover . ■

Definition 2.14 Let be the closed interval [0, 1] in R. Let be

the set obtained from by deleting its "middle third"

. Let

be the set obtained from by deleting its "middle thirds"

and

. In general, define by the equation

∞

.The intersection

is called the Cantor set.

Definition 2.15 A set in a space is perfect in if and only if

is closed and dense in itself; i.e., each point of is an accumulation

9

point of .

The whole space , then, is perfect if and only if it is dense in itself.

In particular, the Cantor set C is perfect.

Lemma 2.16 If is any nonempty open set in a compact totally

disconnected, perfect -space and is any positive integer, then

∪ ⋯ ∪ for some choice of nonempty disjoint open sets

.

Theorem 2.17 Any two totally disconnected, perfect compact metric

spaces are homeomorphic.

Proof. Let be such spaces. Let (), () be sequences of finite

covers of and , respectively, by disjoint open sets, the sets of the

th covers having diameter less than . By using Lemma 2.17 in order

to split sets where necessary, we may assume and have the

same number of elements for each .

Now if { } and { }, then each is

a union of elements of , and each is a union of elements of .

Again, using Lemma 2.17 we can assume and are the union of

the same number of elements of , , respectively, in such a way

that ⊂ if and only if ⊂ . Continue in this fashion,

matching the covers of and for all .

10

Now let

⋯ and

⋯ be the derived

sequences of () and (), respectively. Define → by

. Then is a homeomorphism from to , and it is

easily verified that ∞ → ∞ is also then a homeomorphism. But

∞ is homeomorphic to , and ∞ is homeomorphic to . ■

Theorem 2.18 Every compact metric space is a continuous image

of the Cantor set.

Theorem 2.19 The continuous image of a compact metric space in a

Hausdorff space is metrizable.

11

3. Peano space and Hahn-Mazurkiewicz

Theorem

Here we give a topological characterization of those spaces which are

continuous images of the unit interval I .

we explore some of Peano space properties. We also introduce relation

between Peano space and Hahn-Mazurkiewicz theorem.

Definition 3.1 A Peano space is a compact, connected, locally

connected metric space.

The next three results are directed specifically at the proof of the

Hahn-Mazurkiewicz theorem, which characterizes the continuous images

of I as precisely the Peano spaces.

Theorem 3.2 Every Peano space is path connected.

Suppose and are points in a Peano space . Using Theorem

2.10, there is a simple chain ⋯ of open connected sets of

diameter less than 1 from to . About each point of there is an

open connected set of diameter less than

whose closure is

contained in , and if we can arrange that ⊂ also.

Do this for each ⋯ We wish now to obtain a simple chain of

12

such sets from to .

Pick ∩ for ⋯ and for each ⋯(with

and ) find a simple chain of the sets from to in

We cannot simply join these together to get a simple chain from

to , because of doubling back, but we can obtain the desired simple

chain as follows : take all elements of the first chain (from to ) up

to and including the first one meeting some element of the second

chain (from to ), then omit all elements of the first chain after

and all elements of the second chain before . Repeat this at all other

intersections.

The result, then, is a chain ⋯ of open connected sets of

diameter less than

such that for each ⊂ for some . Now

continue this process, obtaining a simple chain of open connected sets of

diameter less than whose closures lie in elements of the previous

chain for each

For each let be the union of the closures of the elements of

the th chain. Then ∩ is a compact, connected metric space

containing and . We have finished if we show that no points other

than and are noncut points, since then is a path by Theorem 2.12.

Let ∈ For given at most one or two links of the th

chain contain Let be the union of all the links preceding these,

the union of all the links following these. Then

13

∞

∩ and

∞

∩

form a separation of into disjoint, nonempty open sets. Thus

is a cut point of ■

The proof just given can easily be modified to show that every open

connected subset of a Peano space is path connected. We will use this

fact a little later, in the proof of Theorem 3.4.

Lemma 3.3 A compact locally connected metric space is "uniformly

locally connected"; that is, for any , there is some such

that whenever then x and y both lie in some connected

subset of of dimeter less than .

Given cover by open connected neighborhoods of

diameter less than . Reduce this to a finite subcover ⋯ and

let be a Lebesgue number for this cover. Then if both

and belong to some . ■

Theorem 3.4 A Peano space is uniformly locally path connected;

for each , there is a such that whenever , then

x and y are joined by a path of diameter less than .

First, is uniformly locally connected, by Lemma 3..3. Thus if

is given, there is a such that if then and

14

lie together in a connected set of diameter less than Each

has an open connected neighborhood of diameter less than

Then ∪ is an open connected subset of , and hence, is

path connected. Thus, if then and lie in a path

connected subset of diameter less than . ■

We are now ready to prove the Hahn-Mazurkiewicz theorem,

classifying the continuous images of the unit interval as the Peano

spaces. Proving that continuous images of I have the properties of a

Peano space is no trouble ; all the necessary theorems are already at

hand. But to prove the converse is significantly more difficult. The basic

idea is that given any Peano space , there is a continuous map of the

Cantor set onto by Theorem 2.19 and, using the small paths in

provided by the previous theorem, we can extend this map to the whole

unit interval.

Theorem 3.5 (Hahn and Mazurkiewicz) A Hausdorff space is a

continuous image of the unit interval I if and only if it is a Peano

space.

Let be a continuous map of I onto . By Theorem 2.19 and

Theorem 2.13, must be compact and metrizable. Moreover, is the

continuous image of a connected space and a quotient (in fact, a closed,

continuous image) of a locally connected space, so has these

properties itself. Thus, is a Peano space.

15

Now suppose is any Peano space. Recall C is the Cantor set in I ,

with ⋯ being the intervals in I C ordered by size, and for

intervals of the same size, from left to right. Let be a continuous map

of C onto . Our problem is to extend continuously over each

Now and are already defined. If

, define for each Now for each ,

by Theorem 3.4 we can find such that in ⇒

and are joined by a path of diameter less than But, because

→ is uniformly continuous, for each we can find

such that in ⇒

Only finitely many intervals have length greater than or

equal to and for each such extend to by letting its values

run over any path from to in . Then intervals

will have length ≤ For these intervals we have

so that Then and are

joined by a path of diameter less than . Extend to by letting

its values run over any path of diameter less than between and

. In general, for the intervals such that ≤ , we

can let the values of on run over a path of diameter less than

between and . ■

16

References

1. James R. Munkres (2000), Topology. Prentice Hall, Inc.

2. Stephen Willard (2004), General Topology. Dover Publications, Inc. C

3. John G. Hocking and Gail S. Young (1988), Topology. Dover

Publications, Inc.

4. 노영순 (2003), 위상수학 기본. 교우사.

5. Space-filling curve (http://en.wikipedia.org) on Wikipedia