Metric analogs, by Paul R. Meyer

Topology Atlas Document # taic-10.htm | Production Editor: R. Flagg

Topology Atlas Invited Contributions, vol. 1, issue 5 (1996), 82.

Metric analogs
by

Paul R. Meyer

(City University of New York, Lehman College, Bronx, NY 10468, U.S.A.)

A (pseudo-) metric analog of a topological space $X$ is defined to be a (pseudo-) metric space $M$ together with an open quotient map $q:M --> X$ satisfying certain mapping properties. These properties ensure that the homotopy properties of $M$ and $X$ are closely related. Metric analogs are useful tools in the context of the topological space approach to digital topology (see, e.g., [K 1996] for more on digital topology), because they can reduce homotopy problems in digital topology to the more familiar realm of homotopy of metric spaces.

All of the spaces that arise in this context do have metric analogs, but the interesting question of characterizing the class of spaces which have metric analogs has not been fully answered. A partial answer [KKM, Thm. 8]: Each $T_0$ $\sigma$-Alexandroff space has a metric analog, and each space with a metric analog is first countable. However, we don't know how to characterize this class. Since a topology is $\sigma$-Alexandroff iff it has a $\sigma$-interior-preserving base iff it arises from a non-Archimedean quasimetric, this and related problems can be reformulated in terms of concepts in [FL] involving quasi-uniformities.

Metric analogs were introduced by Kong and Khalimsky in [KK] as an outgrowth of Kong's construction of a digital fundamental group [K 1989] and his earlier work on polyhedral and continuous analogs, going back to his Oxford thesis (see [KRR] and earlier refereces there).

References

[FL] P. Fletcher and W.F.Lindgren, Quasi-uniform spaces, M.Dekker, NY, 1982.

[K 1989] T.Y.Kong, A digital fundamental group, Computers and Graphics, 13 (1989) 159-166.

[K 1996] T.Y.Kong, Digital topology, Topology Atlas, Invited Contributions vol. 1, issue 3, 1996, pp. 37-38.

[KK] T.Y.Kong and E.Khalimsky, Polyhedral analogs of locally finite topological spaces, in: R.M.Shortt, Ed., General Topology and Appl: Proc. 1988 Northeast Conf., M.Dekker, NY, 1990, 153-164.

[KKM] T.Y.Kong, R. D. Kopperman, P.R.Meyer (1991). Which Spaces have Metric Analogs?, in: S.J.Andima, et al., Ed., General Topology and Appl.: Fifth North-east Conf., Lecture Notes 134, M.Dekker, NY, 1991, pp. 209-215.

[KRR] T.Y.Kong, A.W.Roscoe, and A.Rosenfeld, Concepts of digital topology, Topology and its Applications, 46 (1992) 219-262.

Received by the editors: February 8, 1996.