Re: The sphere is not plane

David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> writes:

On 20 Sep 2006 09:01:52 -0400, lrudolph@xxxxxxxxx (Lee Rudolph) wrote:
....
This proof has the possible advantage over others that it shows,
not only that there is no homeomorphism from S^2 into C, but that
in fact there is no local homeomorphism from S^2 into C.

Meaning what?

Yeah, I realized about 1 second after posting that that phrase
was, at best, ambiguous. I had been going to say "submersion"
but thought *that* would be too high falutin' (and probably has
"smooth" as part of its definition). Here's a better version:
"This proof has the possible advantage over others that it shows,
not only that there is no homeomorphism from S^2 into C, but that
in fact there is no map f from S^2 into C with the property
that each point P of S^2 has a neighborhood U(P) such that the
restriction of f to U(P) is a homeomorphism from U(P) into S^2."

Lee Rudolph
.

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