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.