Generalized metric spaces, by Gary Gruenhage

Topology Atlas Document # taic-20.htm | Production Editor: R. Flagg

Topology Atlas Invited Contributions, vol. 2, issue 1 (1996), 3.

Generalized metric spaces
by

Gary Gruenhage

(Auburn University, Auburn, AL 36830, U.S.A.)

Generalized metric spaces can refer either to classes of spaces that possess some of the global structure of metrizable spaces, or classes defined by generalizing the notion of a metric in some way. Such classes have a history nearly as old as topology itself: omega_mu metrizable spaces, which can be considered a natural higher cardinal version of metrizable spaces, were studied by Hausdorff in 1914, and semi-metric spaces, where the triangle inequality is dropped, by Frechet and Niemytskii in the 20's. Also in the 20's R.L. Moore defined a non-metrizable "Moore space". But the real flowering of the area came after the Bing-Nagata- Smirnov metrization theorem in '51, and Michael's study of paracompactness in the late 50's. Approximately from then until the early 70's, many new classes of generalized metrizable spaces were defined and studied...M-spaces, p-spaces, sigma-spaces, stratifiable spaces, etc....and many of the leading researchers of the time were engaged in this activity. They were motivated by a desire to more fully understand metrizable spaces, their images and pre-images under "nice" mappings, as well as to find wider classes of spaces which retained some of the useful properties of metrizable spaces. The study continues today, e.g., with the new classes suggested by the "structuring mechanism" of Collins and Roscoe, the results of Rudin and others on monotonically normal spaces, etc. Classes of generalized metric spaces have been useful in dimension theory, function space theory, topological groups, and other areas.

There are a number of survey papers in this area which may be consulted for further information. I wrote one for the Handbook of Set-theoretic Topology (K. Kunen and J.E. Vaughan, eds., North- Holland, 1984), and surveys by J. Nagata, K. Tamano, and Y. Tanaka appear in the book Topics in General Topology (K. Morita and J. Nagata, eds., North-Holland, 1989). I also wrote a survey of the activity in the area during the period 1984-1991; this appears in the book Recent Progress in General Topology (M. Husek and J. Van Mill, eds., North-Holland, 1992). R. Hodel has written an article on the history of generalized metric spaces which will appear in the forthcoming book on the history of general topology edited by C.E. Aull. Many research articles and still more survey articles can be found in the references of the above papers.

Received by the editors: February 6, 1996.