Agoston M.K. Computer Graphics and Geometric Modelling: Mathematics

Подождите немного. Документ загружается.

5.6.5. Theorem. (The Intermediate Value Theorem) Let X be a topological space

and f:X Æ R a continuous map. Assume that f(x

) < f(x

) for some x

, x

僆 X. If c is

a real number so that f(x

) < c < f(x

), then there is an x 僆 X with f(x) = c.

Proof. See [Eise74].

Deﬁnition. A connected subset of a topological space X that is not properly con-

tained in any connected subset of X is called a component of X.

A more intuitive way to express the notion of component is to say that a compo-

nent is a maximal connected subset.

A simpler notion of connected is:

Deﬁnition. Let X be a topological space. We say that X is path-connected if for any

two points p, q 僆 X, there is a continuous map f: [0,1] Æ X with f(0) = p and f(1) =

q. The map f is called a path from p to q. A maximal path-connected subset of a top-

ological space X is called a path-component of X.

5.6.6. Theorem. Let f:X Æ Y be a continuous map from a path-connected space

onto a space Y. Then Y is path-connected.

Proof. See [Eise74].

5.6.7. Theorem. A path-connected space is connected.

Proof. See [Eise74].

Connected does not imply path-connected in general, so that the notion of path-

connected is stronger. For “nice” spaces however these concepts are identical.

5.6.8. Theorem. A topological manifold is connected if and only if it is path-

connected.

Proof. See [Eise74].

5.7 Homotopy

We have talked about how topology studies properties of spaces invariant under

deformations (rubber sheet geometry). This section studies deformations of

mappings.

Deﬁnition. Let f, g:X Æ Y be continuous maps. A homotopy between f and g is a

continuous map

h: XY¥

[]

Æ01,

5.7 Homotopy 309

such that h(x,0) = f(x) and h(x,1) = g(x) for all x 僆 X. In that case, we shall also say

that f is homotopic to g and write f  g.

If we deﬁne f

:X Æ Y by f

(x) = h(x,t), then we can see that the existence of h is

equivalent with a one-parameter family of maps connecting f and g and we can think

of h as deforming f into g. See Figure 5.12(a).

5.7.1. Example. Consider the maps f, g:D

Æ D

given by f(p) = 0 and g(p) = p.

The map h:D

¥ [0,1] Æ D

deﬁned by h(p,t) = tp is a homotopy between them. In

other words, the identity map of D

is homotopic to a constant map.

5.7.2. Theorem. The homotopy relation  is an equivalence relation on the con-

tinuous maps from one topological space to another.

Proof. We must show that the relation is reﬂexive, symmetric, and transitive.

Reﬂexivity: If f :X Æ Y is a continuous map, then h:X ¥ [0,1] Æ Y deﬁned by h(x,t)

= f(x) is a homotopy between f and f.

Symmetry: Let f, g:X Æ Y be continuous maps and assume that h:X ¥ [0,1] Æ Y is

a homotopy between f and g. Deﬁne k:X ¥ [0,1] Æ Y by k(x,t) = h(x,1-t).

Then k is a homotopy between g and f.

Transitivity: Let f, g, h:X Æ Y be continuous maps and assume that a, b:X ¥ [0,1]

Æ Y are homotopies between f and g and g and h, respectively. Deﬁne

g:X ¥ [0,1] Æ Y by

Then g is a homotopy between f and h.

The theorem is proved.

xt x t t

xt t

,,, ,

,, ,.

()

[]

()

[]

2012

21 121

310 5 Point Set Topology

h(X¥0)

p ¥ [0,1]

p ¥ 0

p ¥ t

p ¥ 1

X¥0

X¥1

X¥[0,1]

h(X¥t)

h(X¥1)

= h(p,t)

(a)

(b)

Figure 5.12. Homotopies between maps.

Deﬁnition. If f:X Æ Y is a continuous map, then the homotopy class of f, denoted

by [f], is the equivalence class of f with respect to . The set of homotopy classes of

maps from X to Y will be denoted by [X,Y].

If X consists of a single point p, then a homotopy between two maps f, g: {p} Æ

Y is just a path in Y from the point y

= f(p) to the point y

= g(p). See Figure 5.12(b).

In particular, it is easy to see that the set of homotopy classes [{p},Y] is in one-to-one

correspondence with the path-components of Y.

Deﬁnition. A continuous map f:X Æ Y is called a homotopy equivalence if there is

a continuous map g:Y Æ X with g f  1

and f g  1

. In this case we shall write

X  Y and say that X and Y have the same homotopy type.

5.7.3. Theorem. Homotopy equivalence is an equivalence relation on topological

spaces.

Proof. This is straightforward.

Since the general homeomorphism problem is much too difﬁcult except in certain

very special cases, a weaker classiﬁcation is based on homotopy equivalence.

Deﬁnition. A space is said to be contractible if it has the homotopy type of a single

point.

5.7.4. Example. The unit disk D

is contractible. To see this we show that it has the

same homotopy type as the point 0. Deﬁne maps f:D

Æ 0 and g: 0 Æ D

by f(p) = 0

and g(0) = 0. Clearly, f g = 1

. Deﬁne h:D

¥ [0,1] Æ D

by h(p,t) = tp. Then h is a

homotopy between g f and the identity map on D

, and we are done. Another way to

state the result is to say that both f and g are homotopy equivalences.

Deﬁnition. A subspace A of a space X is called a retract of X if there exists a con-

tinuous map r:X Æ A with r(a) = a for all a in A. The map r is called a retraction of

X onto A.

If x

is any point in a space X, the constant map r(x) = x

shows that any point

of a space is a retract of the space. A less trivial example is

5.7.5. Example. The unit circle in the plane is a retract of the cylinder

(5.10)

because we have the retraction r(x,y,z) = (x,y,0).

Deﬁnition. Let A be a subspace of a space X. A deformation retraction of X onto A

is a continuous map h:X ¥ I Æ X satisfying

and

h h allxxxA xX,,,, ,01

()

ŒŒ

X =

()

+= Œ

[]

{}

x y z X Y and z,, ,

101

5.7 Homotopy 311

In this case A is called a deformation retract of X.

The argument in Example 5.7.4 also shows that the map f:D

Æ 0 deﬁned by

f(p) = 0 is a deformation retraction of D

onto 0.

5.7.6. Example. The unit circle is a deformation retract of the cylinder X deﬁned

by equation (5.10). To see this simply deﬁne h:X ¥ I Æ X by h((x,y,z),t) = (x,y,(1-t)z).

5.7.7. Theorem. Let A be a subspace of a space X. If A is a deformation retract of

X, then the inclusion map i:A Æ X is a homotopy equivalence. In particular, A and

X have the same homotopy type.

Proof. Let h:X ¥ I Æ X be a deformation retraction of X onto A. Deﬁne f: X Æ A by f(x)

= h(x,1). Since f i=1

and h is homotopy between i f and 1

, we are done.

Intuitively speaking, a subset A is a deformation retract of a space X if we can

shrink X down to A without “cutting” anything. We shall see later (Corollary 7.2.3.3

and Theorem 7.2.3.4) that a circle does not have the same homotopy type as a point.

Therefore, no point of the circle is a deformation retract of the circle. The only way

to “shrink” the circle to a point would be to cut it ﬁrst.

Often it is convenient to talk about “pointed” homotopies, or more generally

“relative homotopies.”

Deﬁnition. The notation f:(X,A) Æ (Y,B) will mean that f is a map from X to Y and

f(A) 債 B.

Deﬁnition. Let f, g:(X,A) Æ (Y,B) be continuous maps. A homotopy between f and

g relative A is a continuous map

such that h(x,0) = f(x), h(x,1) = g(x), and h(a,t) 僆 B for all x 僆 X, a 僆 A, and

t 僆 [0,1]. In that case, we shall also say that f is homotopic to g relative A and write

f 

5.7.8. Theorem. The homotopy relation 

is an equivalence relation on the set of

continuous maps f:(X,A) Æ (Y,B).

Proof. The proof is similar to the proof of Theorem 5.7.2. We just have to be careful

that the homotopies keep sending A to B.

Deﬁnition. The set of homotopy classes of maps f:(X,A) Æ (Y,B) with respect to

the equivalence relation 

will be denoted by [(X,A),(Y,B)].

A natural question to ask at this point is how many homotopy classes of maps

there are between spaces in general and what this number measures. We are not ready

h:XY¥

[]

Æ01,

h allaa aA,, .1

()

=Œ

312 5 Point Set Topology

to answer such a question yet and will have to wait until Chapter 7, but the reader

may appreciate one glimpse into the future. Given a map

the degree of f is, intuitively, the number of times that f winds the circle around itself.

(In Section 7.5.1 we shall give another deﬁnition of the degree of f.) Deﬁne

by f

(cosq,sinq) = (cosnq,sinnq). Then f

has degree n. It turns out that all maps of

the circle to itself are homotopic to one of these maps and two maps are homotopic

only if they have the same degree, so that there is a bijection between the homotopy

classes of maps of the circle to itself and the integers.

5.8 Constructing Continuous Functions

There are many situations where one wants to deﬁne continuous functions on a

topological space satisfying certain properties. This brief section describes two very

fundamental theorems that deal with the existence of certain functions, which in turn

can be used to construct many other functions. We shall give one application having

to do with the existence of partitions of unity at the end of the section.

For a topological space to have the continuous functions we want it needs

to satisfy a special property. It is worth isolating this property and giving it a

name.

Deﬁnition. A topological space X is said to be normal if, given two disjoint

closed sets A and B in X, there exist disjoint open sets containing A and B,

respectively.

The condition that a space be normal is somewhat technical, like being Hausdorff,

but fortunately the spaces of interest to us satisfy this property.

5.8.1. Theorem.

(1) Any metrizable space is normal.

(2) Any compact Hausdorff space is normal.

Proof. See [Eise74].

5.8.2. Theorem. (The Urysohn Lemma) Let X be a normal space and assume that

A and B are two closed subsets of X. Then there exists a continuous function f:X Æ

[0,1] such that f takes the value 0 on A and 1 on B.

Proof. See [Jäni84].

:SS

f:SS

Æ ,

5.8 Constructing Continuous Functions 313

5.8.3. Theorem. (Tietze Extension Lemma) Let X be a normal space. Then any

continuous function f:A Æ [a,b] or f: A Æ R deﬁned on a closed subset A of X can

be extended to a continuous function F:X Æ [a,b] or F :X Æ R, respectively.

Proof. See [Jäni84].

The next concept enables one to localize problems and will be used in later

chapters.

Deﬁnition. A partition of unity on a topological space X is a collection F of contin-

uous real-valued functions satisfying the following:

(1) For all j 僆 F and x 僆 X, 0 £j(x) £ 1.

(2) Every point in X has a neighborhood on which all but a ﬁnite number of func-

tions in F vanish.

(3) For every x in X

(Note that by condition (2) this is a ﬁnite sum for each x.)

If x is a cover of X, we say that the partition of unity F is subordinate to x if each

function in F vanishes outside some set in x.

5.8.4. Example. Deﬁne functions

and

(x) = b(x - n).

It is easy to check that the collection of functions b

(x) is a partition of unity on R.

See Figure 5.13. This partition of unity is subordinate to the open cover

b x x for x

x for x

elsewhere

()

=+ Œ-

[]

=- + Œ

[]

110

101

,,,

, : RRÆ

()

314 5 Point Set Topology

n–1

(x) b

n+1

(x)

n–3 n–2 n–1 n+1 n+2 n+3n

Figure 5.13. A partition of unity

for R.

of R.

It would be good to know what condition on a topological space guarantees the

existence of partitions of unity.

Deﬁnition. A cover of a space is locally ﬁnite if every point in the space has a neigh-

borhood that meets only ﬁnitely many elements of the cover. A Hausdorff space is said

to be paracompact if every open cover admits a locally ﬁnite subcover.

5.8.5. Lemma. Every paracompact space is normal.

Proof. See [Jäni84].

5.8.6. Theorem. A Hausdorff space is paracompact if and only if every open cover

admits a partition of unity subordinate to it.

Proof. The only hard part is showing that paracompact implies the existence of the

stated partitions of unity. Because of Lemma 5.8.5 one can use Urysohn’s lemma to

construct the desired partition of unity. See [Jäni84].

The next obvious question is: which spaces are paracompact?

5.8.7. Theorem. The following types of topological spaces are paracompact:

(1) Compact Hausdorff spaces

(2) Topological manifolds

(3) Metrizable spaces

Proof. Part (1) is trivial. Part (2) is also not hard. For (3) see [Schu68].

5.9 The Topology of P

Projective space P

is not only one of the really important spaces in mathematics but

it also serves as an excellent example of a nontrivial topological space. This section

looks at its purely topological properties. We shall return to it later in Chapter 8 to

look at its manifold properties and again in Chapter 10 where its algebraic properties

come to the fore.

Recall the (set theoretic) deﬁnition of P

given in Section 3.4, namely,

(5.11)

where ~ is the equivalence relation on R

n+1

- 0 deﬁned by p ~ cp, for c π 0. In Chapter

3 we did not say anything about its topology, but actually, when we talk about P

a topological space, we always assume that it has been given the quotient space top-

ology that is deﬁned by equation (5.11). What does this space really “look” like top-

PR 0

()

nn n-+

()

{}

11, Z

5.9 The Topology of P

315

ologically? There are quite a few different deﬁnitions that all lead to the same space

(up to homeomorphism). Each gives a little different insight into its structure.

A second deﬁnition of P

is the set of lines through the origin in R

n+1

Justiﬁcation: Except for the fact that the origin is missing, the equivalence class

,...,x

n+1

] is just such a line through the origin, so that there is a natural one-

to-one correspondence of points. (The topologies are assumed to match under this

correspondence.)

A third deﬁnition of P

is the unit sphere S

with antipodal points identiﬁed,

that is,

where p ~-p.

Justiﬁcation: The relation ~ which relates points of S

to their antipodal points is

an equivalence relation, and the map

where D =|(x

,...,x

n+1

)|, is clearly a homeomorphism.

A fourth deﬁnition of P

is the unit disk D

in the plane with antipodal points

on its boundary identiﬁed, that is,

where ~ is induced from the relations p ~-p for p 僆 S

n-1

Justiﬁcation: See Figure 5.14 where the labels and arrows are trying to indicate the

identiﬁcations. The boundary of the upper hemisphere S

is just S

n-1

. It is easy to see

= ~,

xx x

12 1

, ,..., , ,...,

[]

R0 S

nn+

()

~/~,

= /~,

316 5 Point Set Topology

disk with antipodal boundary

points identiﬁed

Figure 5.14. Visualizing the projective plane.

that P

can be thought of as S

/~, where ~ is a restricted version of the equivalence

relation in the third deﬁnition above, namely, p ~-p for p 僆 S

n-1

. Our characteriza-

tion now follows from the observation that the only identiﬁcations that are taking

place are on the boundary of S

and that the interior of S

projects in a one-to-one

fashion onto the interior of the disk D

In the case of the projective plane, there is another well-known identiﬁcation.

A ﬁfth deﬁnition of P

: The projective plane is the union of the Moebius strip and

a disk where we identify their boundaries, which are just circles.

Justiﬁcation: Consider the shaded region in Figure 5.15, which is a “collar” of the

boundary of the unit disk. A little thought should convince the reader that under

the identiﬁcation described in the fourth deﬁnition, this shaded region is just the

Moebius strip.

We now have four different ways of looking at the topological space P

(ﬁve,

in the case of the projective plane). In each case we used a quotient topology of

Euclidean space. Alternatively, one can deﬁne this topology by deﬁning a metric on

Deﬁnition. Let p = [x] and q = [y] be points of P

, where x, y 僆 R

n+1

. Deﬁne the

distance between p and q, denoted dist (p,q), by

It is easy to see that dist(p,q) is well deﬁned and does not depend on the repre-

sentatives x and y that are chosen for p and q, respectively. It is just the angle between

the two “lines” p and q. The function dist(p,q) is in fact a metric on P

and makes P

into a metric space.

5.9.1. Example. Let L be the line in R

deﬁned by -x + 2y + 1 = 0. To ﬁnd the

points of P

that are “near” the ideal point L

⬁

associated to the family of lines

parallel to L.

Solution. Let p = [X,Y,Z] be any point of P

. Since L

⬁

= [-2,-1,0], a simple com-

putation using the deﬁnition of distance shows that the distance d between the two

points satisﬁes

dist d where d and dpq

,, .

()

∑

££cos 0 2p

5.9 The Topology of P

317

Moebius

strip

disk

Disk glued to Moebius strip

along their boundary

Figure 5.15. Another way to visualize the projective plane.

If p corresponds to a real point (x,y), that is, p = [x,y,1], the only way that d will go

to zero is if x and y both get arbitrarily large and (x,y) is near the line y - 2x = 0. This

follows from the Cauchy-Schwarz inequality. Similarly, the only ideal points close to

⬁

are points p = [X,Y,0] with X close to -2 and Y close to -1.

5.9.2. Theorem. N-dimensional projective space P

is a compact, connected,

metrizable topological manifold.

Proof. The compactness and connectedness follows from Lemma 5.3.18 and Theo-

rems 5.4.6 and 5.5.2 using the third and fourth deﬁnition of P

. We postpone showing

that P

is a manifold to Section 8.13, where we will in fact show that it is a differen-

tiable manifold.

Finally, we note that any hyperplane in P

is homeomorphic to P

n-1

. In particu-

lar, the subspace of ideal points is homeomorphic to P

n-1

5.10 EXERCISES

Section 5.2

5.2.1. Prove that every metric on a ﬁnite set is the discrete metric.

5.2.2. Prove that if a vector space V has an inner product <,>, then the function

deﬁnes a metric on V.

5.2.3. Prove that equation (5.1) deﬁnes an inner product on C

([0,1]).

5.2.4. Prove that the function d

deﬁned by equation (5.3) deﬁnes a metric on C

([0,1]).

5.2.5. Prove that the function d

⬁

deﬁned by equation (5.4) deﬁnes a metric on C

([0,1]).

5.2.6. Show that the metrics d

and d

⬁

on C

([0,1]) deﬁned by equations (5.3) and (5.4), respec-

tively, are not equivalent metrics.

5.2.7. Let (X,d) be a metric space. Prove that the function d* deﬁned by equation (5.5) is a

bounded metric on X.

5.2.8. Consider the sequence of functions f

(x) = x

on [0,1]. Show that this sequence of func-

tions converges in a pointwise fashion but not uniformly. Note also that although each

function is continuous, the limit function g(x) is not. Describe g(x).

5.2.9. Show that the rational numbers are not complete by giving an example of a Cauchy

sequence of rational numbers that does not converge to a rational number.

5.2.10. Sometimes one does not quite have a metric on a space.

d uv uv vuvu,,

()

==<-->

cos d

XYZ

()

5,,

318 5 Point Set Topology