Which book (on topology) gives the most complete, yet accessible, account of the Hausdorff metric? the fuzzy metric? the cone metric? the probablistic metric? and so on?

Somebody once gave me a photocopy of a few pages containing a discussion of the Hausdorff metric; it was probably chapter 10 of some book, and it developed the Hausdorff metric axiomatically and derived its properties in a systematic fashion, but he doesn't remember where he took those pages, nor was the title mentioned in the header or the footer.

And what about such other metric spaces, like the cone metric spaces, the fuzzy metric space, and the probablistic metric spaces? I've looked up several books on topology, including Munkres, Simmons, and Lipschitz, but haven't found a discussion of these matters; Kelly gives an account of the Hausdorff metric but only in the exercises.

I have also checked E. T. Copson text on metric spaces and Walter Rudin and Apostol's texts on mathematical analysis. And, as far as I can remember, H. L. Royden doesn't cover these topics either.

  • $\begingroup$ I go along with the idea that an Introduction to the Hausdorff metric and its relation with fractals should be in all undergraduate maths courses. One reason is that the term"fractal" is part of public discourse, and so students should be aware of the maths behind it. I gave such a course in the 1990s at Bangor, in the 2nd year. Without developing metric space theory I just used the notation $d(x,y)$, gave the rules and key examples: Eucidean space, fractal space, and some spaces of functions. $\endgroup$ – Ronnie Brown Aug 16 '13 at 13:46
  • $\begingroup$ You might do better with a book such as Fractals : form, chance, and dimension, Benoit B. Mandelbrot (1977) which gives a lot of context. $\endgroup$ – Ronnie Brown Aug 16 '13 at 17:25
  • $\begingroup$ Actually I need this sort of material in connection with fixed point theory, especially in the discussion of the multivalued maps. $\endgroup$ – Saaqib Mahmood Aug 18 '13 at 5:35
  • $\begingroup$ Point Sets by Eduard Cech (1969) has one of the nicest treatments of the Hausdorff metric that I know of from a pure mathematics viewpoint. Oops, I didn't realize this was a 3+ year old question! Well, maybe someone else will benefit . . . $\endgroup$ – Dave L. Renfro Nov 29 '16 at 20:35
  • $\begingroup$ @DaveL.Renfro thank you so much for answering my question. $\endgroup$ – Saaqib Mahmood Nov 30 '16 at 12:52

You can check out Encyclopedia of Distances by Deza and Deza.

  • $\begingroup$ Thank you, but is the presentation there going to be elementary enough? I wonder if it is just a compendium of facts, which doesn't systematically discuss why and how such and such a result holds. $\endgroup$ – Saaqib Mahmood Aug 16 '13 at 10:54
  • $\begingroup$ @Saaqib: It may be that the only way to get a book to your liking is to write it yourself! Isn't there a fable about Brer Rabbit of this type? In the meanwhile, Chaos and fractals : new frontiers of science / Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, could be useful. $\endgroup$ – Ronnie Brown Aug 24 '13 at 16:27

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