14 0. Prequel

0.6.2. Every ∂-manifold is an ANR.

0.6.3. Every polyhedron is an ANR.

0.6.4. Give an example of an ANR that is not homeomorphic to a poly-

hedron.

0.7. Dimension theory

Munkres (2000, §50) (or (1975, §7-9)) contains most, but not quite all, of

what the reader needs to know about dimension theory. This section fills in

the necessary background. All topological spaces considered in this book are

separable metric spaces, so some of the technical complications of dimension

theory for more general spaces can be avoided.

Let U be an open cover of the space X. Say order(U) ≤ k +1 if no point

of X lies in more than k +1 of the elements of U. The statement dim X ≤ k

means that for every open cover U of X there exists an open cover V of X

such that V is a refinement of U and order(V) ≤ k + 1.

A function f : X → Y defined on a metric space is an -mapping if

diam f

−1(y)

for every y ∈ Y . When dim X ≤ k and 0 is given,

it is a fairly simple matter to use the nerve of a cover to construct an -

mapping of X into a k-dimensional polyhedron (Hurewicz and Wallman,

1948, pages 67–70). Alexandroff proved that, by modifying the polyhedron,

this mapping can be made onto and that the existence of -mappings into

k-dimensional polyhedra characterizes compacta of dimension ≤ k. The

following result appears as Theorem V10 in (Hurewicz and Wallman, 1948).

Theorem 0.7.1 (Alexandroff). A compact metric space X has dimension

≤ k if and only if for every 0 there exists an -mapping of X onto a

polyhedron of dimension ≤ k.

Sketch of proof. When X admits an -mapping φ onto a k-dimensional

polyhedron P , one can compute η 0 such that diam

φ−1(x)

for all

x ∈ P . By subdividing P and carefully thickening the interiors of the

various simplices of the subdivision, one can produce an open cover V of P

with order(V) ≤ k + 1 and with diam V η for all V ∈ V. Then

U =

{φ−1(V

) | V ∈ V}

is a small mesh open cover of X of order k + 1.

If X is a subset of a PL manifold, the polyhedron can be realized as a

subcomplex of the manifold. The following corollary illustrates how we will

make use of the dimension of an arbitrary compactum and the proof is an

early indication of how local connectivity will be used in this book.