Re: Cut a point into two - topological?

Lee Rudolph wrote:
Hero writes:
The incenter and the outcenter of a triangel with equal sides are
identical. In dynamical geometry i can deform the triangel and see two
different centers emerge from one point (location) and by moving back
they melt again into one in this place. (In numbers we can add 2 and 5
into 7, and we can split 7 into two different primes).
Topology is considering in geometry points and (,,open") sets of
points, especially sets of ,,connected"points. So the opposite term of
a ,,cut", a dissection, separation is often heard about, also sometimes
a ,,cut and paste" or ,,cut and glue"-operation is mentioned. The
opposite to a union of a subset and it's complement in set theory
divides a set into two subsets with its intersection equal to the
empty set.
How does topology defines a ,,cut"?
Can one cut a point into two?

There are several different, mutually inconsistent, definitions
of "cut and paste" (_schneiden und kleben_, nicht wahr?), suited
for different purposes. There is also (to go back to your "dynamical
geometry" example) a notion, in old-fashioned algebraic geometry
(though it's coming back into fashion), of "infinitely near points".
So one way to "cut a point into two" is to displace on (of two or more)
"infinitely near points" from the other(s).


That "infinitely near points" is not, what i'm looking for ( as these
are either in the same spot (location) or they have a finite
distance). It 's more about getting more insight into the other
"several different, mutually inconsistent, definitions of "cut and
paste"...., suited for different purposes."

With friendly greetings
Hero
PS
(_schneiden und kleben_, nicht wahr?)
Ja, das ist gutes Deutsch.


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