Table Of ContentDimension Theory
KEIO NAGAMl
DEPARTMENT OF MATHEMATICS
EHIME UNIVERSITY
MATSUYAMA, JAPAN
with an appendix by Yukihiro Kodama
TOKYO UNIVERSITY OF EDUCATION
TOKYO, JAPAN
ACADEMIC PRESS New York and London 1970
COPYRIGHT 0 1970, BY ACADEMICPR ESS, INC.
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Dimension theory developed from an initial effort to give a rigorous
definition of dimension for the line and the plane. By 1941, as can be seen
from Hurewicz and Wallman’s “Dimension Theory,” classical dimension
theory for separable metric spaces had been largely completed. The
highlights of this theory include Brouwer’s invariance of domain theorem,
Lebesgue’s covering theorem, and the fundamental equalities connecting
the three basic dimension functions: in our notation, ind = dim = Ind.
In spite of many efforts to extend dimension theory to nonseparable
spaces there was almost no progress before DieudonnC and Stone estab-
lished a foundation for the theory of open coverings. This theory is given in
Chapter 1. In 1952-1954 Katttov and Morita used this theory to establish
the equality dim = Tnd for general metric spaces. Since that time, many
other interesting results have been discovered, some of which are new
even for separable metric spaces.
It seems difficult at the present stage to go beyond metric spaces and
still retain the equality dim = Ind. Then are we at a dead end? The answer
is no. The gaps between the basic dimension functions, far from being
obstructions, are very interesting in themselves. The author thinks of the
present status of dimension theory as a symphony having basic rhythms,
some of which are ind, dim, Ind, and metric-dependent dimension func-
tions. Hence, in general, they play different parts. In this book the reader
will look at some of the ‘‘music’’ that appears. And a study of the gaps has
this advantage: If we have enough knowledge about the gaps, then by
avoiding them carefully it may be easier to find a new class of spaces
where a “harmonious” dimension theory could be developed.
ix
X Preface
The aim of this book is thus to study the theory not only for metric
spaces but also for nonmetric spaces. So the present book has different
features than Nagata’s “Modern Dimension Theory,” in which the main
concern is with metric spaces. Another characteristic of the present book
is the inclusion of many examples. Students who have only an elementary
knowledge of general topology will be able to read the book (except the
appendix) without difficulty. We expect the reader to be familiar with
metric spaces, compact spaces, the Hilbert cube, and Tietze’s extension
theorem for normal spaces. Chapter 1 is designed to be used as a modern
textbook for a second course in general topology. Except for Sections 5
and 7 this chapter may also be used as an introduction to algebraic
topology. If the reader would like to have an outline of modern dimension
theory rapidly, the author recommends Sections 1,2,3, and 6 in Chapter 1,
Sections 8 and 9 in Chapter 2, and Section 12 in Chapter 3. The book is
based on lectures given at Duke University, 1963-1965, and at the Univer-
sity of Pittsburgh, 1968-1969.
ACKNOWLEDGMENTS
Professor Y. Kodama has given us an exciting present: an appendix
which will lead the reader rapidly to the present state of algebraic dimen-
sion theory. The so-called logarithmic law for dimension, in particular,
requires algebraic treatment.
The author wishes to express his sincere thanks to Professors J. Nagata
and J. H. Roberts who gave constant encouragement, to Professors Y.
Kodama and A. Okuyama who carefully read the proof and made valu-
able suggestions, to Professors R. Hodel and J. Vaughan who corrected
a part of the manuscript in its English version, and to the staff of Adademic
Press for its helpful cooperation.
xi
Chapter 1
THEORY OF OPEN COVERINGS
Throughout this book all spaces are TI-spaces,i .e., spaces in which
every point is a closed set. Our main concern is with normal spaces or
spaces with stronger conditions than normality. It is also to be noted that
coverings are not necessarily assumed to be open. Closed coverings often
play an important role in this book. The index i always ranges countably,
and usually it ranges over the positive integers.
This chapter is mainly concerned with the theory of open coverings
and its basic applications. The normality of open coverings is a powerful
tool and often reflects global properties of a space. Many important
topological problems can be reduced to the question of the normality
of certain types of open coverings. This technique is used throughout the
book. Section 1 is a study of simplicia1 complexes having two kinds of
topologies: the weak topology and the metric topology. In Section 2 the
reader will see the problem of the normality of a given open covering of
a space translated to the corresponding problem on a simplicial complex
with the metric topology. Since it is easy and convenient to handle metric
spaces, this technique changes difficult problems to easy ones. After
establishing several criteria for the normality of open coverings in
Section 2, we apply them in Section's 3, 4, and 5 to the modernized theory
of extensors and absolute retracts and to metrization theorems. In
Section 6 inverse limits and the bridge theorem are studied. The bridge
theorem (Corollary 6-8) will play an important role in Section 28 in
treating the dimension of the complement of a disjoint sum. The theory
of inverse limits is also one of the leading ideas throughout the book. The
advantage is to change a complicated problem to a simple one about each
1
2 1 Theory of Open Coverings
coordinate space in the inverse limiting system. Section 7 is a study of
totally normal spaces, a class of spaces well-designed for the large induc-
tive dimension. Total normality is a generalization of the classical concept
of perfect normality.
1. Complexes
1-1. DEFINITIOLNet . K = {s,} be a set and < be a relation defined for
some pairs of elements of K such that :
i. If sp < s, and sy < sp, then s, < s ,
ii. 4 (the empty set) is in K
iii. For any s, E K, 4 < s ,
Then K with the relation < is an abstract complex. If sg < s,, then sp
is a face of s,; furthermore, if sp # s,, sp # 4, then sp is a proper face of s,.
Let L be a subset of K. If the statements s, E L and sp < s, imply that
sg E L, then L is a subcomplex of K.
Let P be a set and K(P) the set of all finite subsets of P. For any pair of
elements of K(P), say s, and sp, we define s, < sp if and only if s, is a subset
of ss . The abstract complex K(P) thus obtained is a full complex spanned
by P, and P is the vertex set of K(P). A point u of P is a riertex of K(P). If
s, E K(P) and v E s,, then u is a vertex of s, . If
s={uo, ...) om)
then s is spanned by uo, ..., urn and the number m is the combinatorial
dimension (or merely the dimension) of s, denoted by dim s. If L is a sub-
complex of K(P), then L is a simplicial complex. An element of L is a
simplex of L. An n-simplex is a simplex whose dimension is n. If we set
Q={V:ZIESS, EL}
then Q is the vertex sef of L and elements of Q are vertices of L. The
combinatorial dimension of L, dim L, is
max{n: dim s = n, s EL)
dim L may be infinite. Let I Q I be the power of Q. When I Q I < 03,
I Q I = 03, or I Q 1 KO,L is, respectively, a finite, an infinite, or a
countable simplicial complex. If each vertex of L is not a face of an infinite
number of simplices of L, then L is star-jinite. The n-skeleton of L is the
subcomplex
(s: s E L, dim s n}
1 Complexes 3
Let P be a set with I P I > n. Then the n-full complex spanned by P
is the n-skeleton of the full complex spanned by P.
1-2. DEFINITIOLNe.t L be a simplicia1 complex and Q the vertex set
of L. Let I L I be the aggregate of all forms
1
x = A" * 11, s E L
VES
where the &'s are nonnegative real numbers and C, 2" = 1. Let I s I be
the aggregate of all forms: CUESA, . u in I L 1, where s EL. Let
po = (O,O, ...),pI= (1,0, ...), p2 = (0, 1/2,0, ...), etc., be points of the
Hilbert cube I". Then any finite points of them are in general position.
Let en, a Euclidean n-simplex, be the closed convex hull spanned by
po,pl, ... ,pn.T hen every point p of en is expressed as Ai -piw here
the Ai's are the barycentric coordinates? with respect to po, . .. , pn. For
any simplex s spanned by uo , . . . , u,, , define the transformation cps: I s I + en
as follows :
Let us introduce a topology into J L 1 in such a way that a subset G of
1 L [ is open if and only if, for each simplex SEL,w here dims= n,
q,(Gn I s I) is open in en.T hen I L I is evidently a topological space, i.e.,
a T,-space. This topology is called the weak topology of I L I. I L I with
the weak topology is denoted by I L ,,,I and is called a (simplicial) poly-
hedron with the weak topology. If u is a vertex of L, 1 . u in I L I is called a
vertex of I L I and is expressed simply as u.
Let us define another topology for I L I. Every point x of I L I can be
written as C,,, A, u, where all but a finite number of 2"'s are zero. Take
* -
two points x = C Au u and y = X pa . u expressed in this manner. Set
P(X9 v)= [(C:n, - Po) 2 1 112
Then p is a metric on I L I. The topology of I L I defined by this metric
is called the metric topology of I L I. I L I with the metric topology is
denoted by I L lmet and is called a (simplicial) polyhedron with the metric
topology. I L Iw and I L lme, are geometric realizations of L. In the sequel
1 L I stands for 1 L ,,,I or I L Imet. Sometimes a polyhedron I L 1 implies a
pair (I L 1, L). A subpolyhedron of I L I is a subspace I M 1 of I L I corre-
sponding to a subcomplex M of L. When L is full or n-full, I L I is called
7 Compare Hurewicz and Walliiian [I, p. 70, footnote *I.
4 1 Theory of Open Coverings
full or n-full, respectively. For any n-simplex s E L, I s I with either the
weak topology or the metric topology is homeomorphic to a Euclidean
n-simplex en. I s I with this unique topology is called an n-simplex again.
Let x be a point of 1 L 1 with x = XusQ I, . v. Then the Lo’s are barycentric
coordinates of x and each I, is the weight on u of x. Let u be a vertex of
1 L 1. Set the star of ti:
St(v) = {x: the weight on zi of x is positive}
1-3. REMARKL.e t L be a simplicia1 complex, X a space, and f a trans-
formation of I L Iw to X. Then f is continuous if and only if for every
simplex s E L,fl I s I, the restriction off to 1 s I, is continuous. Therefore,
the identity transformation g: 1 L Iw + I L lmet is continuous. In other
words the weak topology is stronger (!) than the metric topology. If L
is finite or star-finite, then g is a homeomorphism.
1-4. DEFINITIOLNet. X be a set and 9 = {Ga:c t EA) a collection of
subsets of X. Let K(A) be the full complex spanned by A and let
N(9) = {s E K(A): n{G,: a ES} # 4)
Then N(9) is a subcomplex of K(A) and is called the nerve of 9.A geo-
metric realization of N(9), I N(9) ,,, I or [ N(9) lmefr is a geometric nerve,
or simply a nerve, of 9.
Let X be a topological space and 9 = {Ga:c t E A} an open covering of X.
9 is locallyjnite if every point x in X has a neighborhood which meets at
most a finite number of elements of 9.I t is to be noted that a neighbor-
hood is not necessarily open in this book.
1-5. THEORELMet . X be a topological space and 9 = {Ga:a E A} a
locally-finite open covering of X such that for every a there exists a
continuous nonnegative function fad efined on X with G, = {x:.fu(x>) O},?
then the transformation f defined by
is a mappingt of X to the nerve I N(9) I,,,
,
t This type of set is a cozero set. A set is a zero set if its complement is a cozero set.
:I A mapping is always a continuous transformation in this book.
1 Complexes 5
PROOF. First f is clearly a transformation into I N(9) IW. To prove the
continuity off, let x be an arbitrary point of X and U an open neighbor-
hood of x such that
A’={ct: UnG,#4}
is a finite set of indices. Set
Then L is a finite subcomplex of N(9) whose vertex set is A’. f l U is a
transformation of U to I L I. Since for each y E U
.fh(Y>/ CfP(Y) =f,(Y)/ CfP(Y)
PEA &A’
and the right side is continuous on U, the weight of f(y) on ci E A‘ is
continuous. Thusf 1 U is continuous and hence f is continuous.0
This mapping.fis called the canonical mapping of 9 (with respect to the
fa’s). It is to be noted that, for every a, f - ‘(St(cc)) = G,.
1-6. DEFINITIOAN c.o llection of sets q/ is refined by a collection of sets
9‘- if for any set V E V there exists a set U E @ with V c U. V < 42 is the
abbreviation of this fact.
Let X be a space and 9 an open covering of X. If there exists a poly-
hedron I L I and a mapping f : X I L I such that 9 is refined by
--f
{ f -‘(St(v)): u E the vertex set of I L I}
then 9 is said to be approximated by the polyhedron I L I.?
Let (X,p ) be a metric space with a metric p. Let S,(x) = {y:p (x, y) < E}.
For a subset H of X, S,(H) = u{S,(x): x E H}. If for every E > 0, the
&-covering{ S,(x):x E X} is approximated by a polyhedron, (X,p ) is said
to be approximated by polyhedra.
The study of metric spaces which can be approximated by polyhedra
of dimension 5 n has been one of the most serious problems in dimension
theory. We shall discuss this problem in Chapter 7.
x
TERMINOLOGY. A collection q/ of subsets of a set is point-jnite if
every point x of Xis contained in at most a finite number of elements of %.
.f If 9 is approximated by a polyhedron with the weak topology, then 9 is approxi-
mated by the same polyhedron with the metric topology, because of Remark 1-3.
