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Unformatted text preview: are not equal, in fact, they are disjoint, yet the
Hausdorﬀ distance between them is 0. On the other hand, the proof of positivedeﬁniteness is convincing. Have we found a logical inconsistency that will bring
down all of mathematics, and so all of physics? Will airplanes fall out of the
sky because of this? No: recall we deﬁned the Hausdorﬀ metric on compact
sets, and Q and I are not compact, because neither is closed. FiThe part of the
proof of Prop. 2.2.1 that relies on compactness of the sets will reveal why this
argument cannot be applied to Q and I .
PrxsSol 2.2.2 (a) The Sierpinski carpet consists of 8 copies of itself, each
scaled by r = s = 1/3, without rotation or reﬂection. Following the image of
the lower left corner (0, 0) of the carpet in each small copy of the carpet, we
ﬁnd the translations and consequently the IFS rules in the table below.
(b) For an ǫ-nbhd Oǫ to be simply-connected, we must ﬁll all the holes in O.
The largest is the 1/3 × 1/3 hole in the middle, and the point in that hole
farthest from O is (1/2, 1/2). The distance between th...
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