Fred Lunnon writes:

I must say I find this computation most unconvincing!

I agree. My intention was to be provocative and elicit appropriate incredulity. A fuller (and hopefully more convincing) account follows.

Not only that, but you include an "F" term appertaining to 3-space, when your interval is (presumably) considered embedded in 2-space: while this might be (partially) consistent with a rationalised definition, it is inconsistent with the one you have used.

It's worse than you think. Not only do I append an "F" term, I include infinitely many, since I claim that the set of polyhedral maps from (0,1) to {0,1} (i.e., the set of piecewise constant maps with only finitely many pieces) is most naturally viewed as infinite-dimensional! More precisely, I claim that the natural way to build up the set of such maps uses 2 vertices (0-cells), 6 edges (1-cells), 18 faces (2-cells), 54 3-cells, etc. To compute the "Euler characteristic" of this infinite complex, sum the geometric series 2-6+18-54+... the way Euler would to get 2/(1-(-3)) = 1/2. But an alternate method involves a theorem to the effect that the Euler characteristic of the set of polyhedral maps from polyhedral set S to polyhedral set T is chi(T) to the power of chi(S), which justifies the flimflam about 2 to the power of -1. (Except that I should really say "Euler measure" instead of "Euler characteristic" to make sure nobody gets confused about what I mean.)

In the case of polyhedra [or polytopes generally] the "Euler characteristic" as commonly defined is defective, in that it omits both the polyhedron itself and the null element [of dimension d = -1] .

I agree about the former, but I'm not sure about the latter. One virtue of Stephen Schanuel's way of looking at Euler measure (which counts the polyhedron but not the null element) is that it's additive (and indeed finitely additive): The Euler measure of a disjoint union of sets is the sum of the Euler measures of the sets themselves. This works for all polyhedral sets (not just compact or even bounded ones). An open k-dimensional cell has Euler measure (-1)^k, so V-E+F-... is just the sum of the Euler measures of the "pieces" of our polytope. Meanwhile, a closed k-dimensional cell has Euler measure 1^k = 1. The formula V-E+F=2 (and its generalization to higher dimensions) follows almost trivially from the additivity property: e.g., you can view a cube as either the disjoint union of 8 0-cells, 12 1-cells, and 6 2-cells (with Euler measure 8-12+6) or as a closed cube minus an open cube (with Euler measure 1-(-1)). Of course, some dirt is being swept under the rug here, but isn't mathematics just the art of sweeping ever-more dirt under ever-smaller rugs? In this case, the sweeping involves iteratively defining 1-dimensional Euler measure, 1-dimensional Euler integration, 2-dimensional Euler measure, 2-dimensional Euler integration, etc., proving a Fubini theorem and whatnot as one goes. I think Rota and Klain's book "Introduction to Geometric Probability" has a nice treatment of this (though it may be in some other papers of Rota). Jim Propp