Newsgroups: sci.math
From: hsbrand@cs.vu.nl (HS Brandsma)
Subject: Re: Dimension theory
Date: Tue, 1 Oct 1996 13:15:40 GMT

R.J.Chapman (rjchapma@exeter.ac.uk) wrote:
: It is well-known that any (compact metric) space of dimension d can be
: embedded in R^(2d+1). Also this result is best possible, in that for each d
: there is a space of dimension d which cannot be embedded in R^(2d). The
: example usually given is the d-skeleton of the (2d+2)-dimensional simplex
: (i.e., the simplex with 2d+3 vertices). For d = 1 this is the complete
: graph on five vertices, and it's an easy exercise to prove that this
: cannot be embedded in R^2 by using the Jordan curve theorem. For d >= 2
: the result is harder. All the topology books I've consulted refer
: to a paper by Flores from the 1930s in German. Alas, both the local library
: and my German leave something to be desired. Does anyone know of a more
: recent and accesible reference, or can give an outline of the method of proof.

: Thanks,

: Robin Chapman
: -- 
: Robin J. Chapman		 	
: Department of Mathematics		"... But there are full professors
: University of Exeter, EX4 4QE, UK	 in this place who read nothing
: rjc@maths.exeter.ac.uk             	 but cereal boxes."
: http://www.maths.ex.ac.uk/~rjc/rjc.html	 	Don Delillo--White Noise


Have a look at Engelking's "new" book on Dimension theory:
page 106 exercise 1.11H gives an outline of the proof with
hints. Page 100 gives the proof for n=2.
This is probably also in his old book on Dimension theory, as
chapter 1 is almost literally the same in both books. I haven't checked
that though..
(Title of the new book: theory of dimensions, finite and infinite)

Henno Brandsma