© 1996, Topology Atlas

We mean by a special metric a metric function on a metrizable space $X$ which induces the original topology of $X$ and has a metric property which determines a topological property on $X$. It is well known that a totally bounded metric and a complete metric are special metrics which characterize the separability and the Cech completeness of spaces, respectively.

Especially, special metrics have a great advance in dimension theory. Here, I shall present an introduction of special metrics in dimension theory. The first attempt of this aspect is a well known characteriation of strongly zero-dimensional spaces by means of non-Archimedean metrics due to Hausdorff and de Groot. A metric $d$ on a space $X$ is called a non-Archimedean metric if $d(x,z) <= max (d(x,y), d(y,z))$ for every three points $x$, $y$, and $z$ of $X$. Nagata and Ostrand extended the theorem above to n-dimensional spaces as follows, independently.

A metrizable space $X$ has $dim X <= n$ if and only if it admits a metric $d$ satisfying the condition $(1)_{n}$ :

For every $n+3$ points $x$, $y_1$, ... , $y_{n+2}$ of $X$ there is a pair of distinct indices $i$, $j$ such that $d(y_i, y_j) <= d(x, y_j)$.

Nagata proved the theorem by use of a normal sequence with special property. Using the same technique, he also obtained other special metrics which characterizes n-dimensional spaces and certain infinite dimensional spaces. Hattori furthered Nagata's results and obtained other special metrics characterizing certain infinite dimensional spaces.

The following problem still remains open since de Groot raised it in 1957.

Problem (J.de Groot [G2]). Is it true that a metrizable space $X$ has $dim X <= n$ if and only if $X$ admits a metric $d$ satisfying the condition $(2)_{n}$ :

For every $n+3$ points $x$, $y_1$, ..., $y_{n+2}$ of $X$ there is a triplet of indicies $i,j,k$ with $i$ not equal to $j$ such that $d(y_i, y_j) <= d(x, y_k)$?

The reader is referred to [HN] for the recent progress (including some background information) on special metrics. This survey also contains some topics related to uniform spaces and other areas in general topology. The reader is also referred to the books on dimension theory : [E, Section 4.2] and [N, Sections V.3 and V.4].

References

[E] R. Engelking, Theory of dimensions. Finite and infinite, Heldermann Verlag (1995).

[HN] Y. Hattori and J. Nagata, Special metrics, in M. Husek and J. van Mill eds., Recent Progress in General Topology, North-Holland (1992), 354-367.

[N] J. Nagata, Modern dimension theory (revised and extended edition), Heldermann Verlag (1983).