Dimension and Maps

Abstract

In this chapter, we establish Urysohn’s lemma (Lemma 4.1.2) and the Tietze extension theorem (Theorem 4.1.4) for normal spaces.

Appendices

Notes

The Tietze extension theorem (cf. Theorem 4.1.4), also called the Tietze-Urysohn extension theorem, was first proved for metric spaces by Tietze [104] and then generalized to normal spaces by Urysohn [109].

The notion of a nerve was introduced by Alexandroff in [3]. Theorem 4.4.7 as well as Theorem 4.5.4 are also contained in that paper of Alexandroff.

A variant of $\dim _\varepsilon (X,d)$ introduced by Gromov [44, Sect. I.1] is ${{\mathrm{Widim}}}_\varepsilon (X,d)$ which is defined, for any compact metric space (X, d) and $\varepsilon > 0$, as being the smallest integer n such that there exists an $\varepsilon$-injective continuous map $f :X \rightarrow P$ from X into some n-dimensional polyhedron P (cf. Exercise 4.11). Motivated by a question raised by Gromov [44, p. 334], Gournay [40] and Tsukamoto [107] obtained interesting estimates for ${{\mathrm{Widim}}}_\varepsilon (X,d)$ when X is the $\ell ^p$-ball in $\mathbb {R}^n$ and d is the metric induced by the $\ell ^q$-norm.

Let X and Y be topological spaces that are not both empty. For X and Y compact and metrizable, it may happen that the inequality $\dim (X \times Y) \le \dim (X) +\,\dim (Y)$ in Corollary 4.5.6 is strict. Indeed, in 1930, Pontryagin [91] (see also his survey paper [92, Sect. 11]) gave examples of compact metrizable spaces X and Y with $\dim (X) = \dim (Y) = 2$ but $\dim (X \times Y) = 3$. Actually, the dimension of the product of two compact metrizable spaces can deviate arbitrarily from the sum of the dimension. More precisely, it was proved in the 1980 s by Dranishnikov (see the survey paper [31]) that, given any positive integers n, m, k with $\max (n,m)+1\le k\le n+m$, there exist compact metrizable spaces X and Y satisfying $\dim (X) = n$, $\dim (Y) = m$, and $\dim (X\times Y)=k$. Recall that we always have $\dim (X \times Y) = \dim (X) + \dim (Y)$ if X and Y are polyhedra by Corollary 3.5.10. The inequality $\dim (X \times Y) \le \dim (X) + \dim (Y)$ remains valid when X and Y are both compact Hausdorff or both metrizable (see [77], [33, Th. 3.4.9]). In [77], Morita proved the inequality $\dim (X \times Y) \le \dim (X) +\dim (Y)$ in the case when X and Y are paracompact Hausdorff spaces with Y locally compact (see [79, p. 153]). Recall that every metrizable space is paracompact. By a result of Hurewicz [49], one has $\dim (X \times Y) = \dim (X)\,+\,\dim (Y)$ whenever X is a non-empty compact metrizable space and Y a separable metrizable space with $\dim (Y) = 1$. In this last result, the compactness hypothesis on X cannot be removed. Indeed, Erdös [34] gave an example of a separable metrizable space X such that $\dim (X \times X) = \dim (X) = 1$ (see Sect. 5.1). On the other hand, Wage [114] described a separable metrizable space X and a paracompact Hausdorff space Y such that $\dim (X \times Y) = 1 > \dim (X) + \dim (Y) = 0$. The result of Corollary 4.7.6 (Menger-Nöbeling theorem) is optimal in the sense that for every integer $n \ge 0$ there exists a compact metrizable space X with $\dim (X) = n$ that cannot be embedded in $\mathbb {R}^{2n}$. One can take as X the n-skeleton, i.e., the union of the n-dimensional faces, of a $(2n + 2)$-simplex (see [33, p. 101]). The idea of using Baire’s theorem in order to prove the Menger-Nöbeling embedding theorem is due to Hurewicz.

Exercises

• 4.1 Let X be a metric space. Let A and B be disjoint closed subsets of X. Show that the map $f :X \rightarrow [0,1]$ defined by

  $$ f(x) := \frac{{{\mathrm{dist}}}(x,A)}{{{\mathrm{dist}}}(x,A) + {{\mathrm{dist}}}(x,B)} $$

  is continuous and satisfies $A = f^{-1}(0)$ and $B = f^{-1}(1)$.

• 4.2 Show that in the statement of Lemma 4.1.2 one cannot replace the condition $A \subset f^{-1}(0)$ by the condition $A = f^{-1}(0)$. Hint: consider for example the product space $X = [0,1]^\mathbb {R}$ with $A = \{a\}$ and $B = \{b\}$, where a and b are distinct points in X.

• 4.3 Let X be a topological space. Suppose that for every finite open cover $\alpha$ of X, there exists a partition of unity subordinate to $\alpha$. Show that X is normal.

• 4.4 Let $\alpha = (U_i)_{i \in I}$ be a finite open cover of a topological space X. Let C denote the geometric realization of the nerve of $\alpha$. Suppose that $(f_i)_{i \in I}$ and $(g_i)_{i \in I}$ are partitions of unity subordinate to $\alpha$. Let $t \in [0,1]$. Show that $((1-t)f_i + tg_i)_{i \in I}$ is a partition of unity subordinate to $\alpha$. Deduce that the maps $f :X \rightarrow \vert C \vert$ and $g :X \rightarrow \vert C \vert$ associated with $(f_i)_{i \in I}$ and $(g_i)_{i \in I}$ respectively are homotopic, i.e., there exists a continuous map $H :X \times [0,1] \rightarrow \vert C \vert$ such that $H(x,0) = f(x)$ and $H(x,1) = g(x)$ for all $x \in X$.

• 4.5 Let X and Y be compact metrizable spaces. Show that if $\dim (Y) = 0$, then one has $\dim (X \times Y) = \dim (X)$.

• 4.6 Let X be a non-empty compact metric space. Show that one has $\dim _\varepsilon (X) = 0$ for every $\varepsilon > {{\mathrm{diam}}}(X)$.

• 4.7 Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. The set $X \times Y$ is equipped with the metric d defined by

  $$ d((x_1,y_1),(x_2,y_2)) := \max (d_X(x_1,x_2),d_Y(y_1,y_2)) $$

  for all $(x_1,y_1), (x_2,y_2) \in X \times Y$. Show that one has

  $$ \dim _\varepsilon (X \times Y,d) \le \dim _\varepsilon (X,d_X) + \dim _\varepsilon (Y,d_Y) $$

  for every $\varepsilon > 0$.

• 4.8 Let $d_1$ and $d_2$ be two metrics on a set X. Consider the metric d on X defined by $d := \max (d_1,d_2)$. Show that one has

  $$ \dim _\varepsilon (X,d) \le \dim _\varepsilon (X,d_1) + \dim _\varepsilon (X,d_2) $$

  for every $\varepsilon > 0$.

• 4.9 Let d denote the Euclidean metric on $[0,1]^2$. Compute $\dim _\varepsilon ([0,1]^2,d)$ for every $\varepsilon > 0$.

• 4.10 Let $n \in \mathbb {N}$ and $p \in [1,\infty ]$. Denote by d the metric associated with the p-norm $\Vert \cdot \Vert _p$ on $\mathbb {R}^n$. Show that one has

  $$ \dim _\varepsilon (\mathbb {R}^n,d) = n $$

  for every $\varepsilon > 0$.

• 4.11 Let (X, d) be a compact metric space and let $\varepsilon > 0$.

  • (a) Show that there exist a polyhedron P and an $\varepsilon$-injective continuous map $f :X \rightarrow P$.

  • (b) Let ${{\mathrm{Widim}}}_\varepsilon (X,d)$ denote the smallest integer n such that there exist a polyhedron P with $\dim (P) = n$ and an $\varepsilon$-injective continuous map $f :X \rightarrow P$. Show that one has $\dim _\varepsilon (X,d) \le {{\mathrm{Widim}}}_\varepsilon (X,d) \le 2 \dim _\varepsilon (X,d) + 1$.

  • (c) Determine $\dim _\varepsilon (X,d)$ and ${{\mathrm{Widim}}}_\varepsilon (X,d)$ for every $\varepsilon >0$ when X is the Cantor ternary set and d is the usual metric on $X \subset \mathbb {R}$.