Re: Cut a point into two - topological?

Hero wrote:
So a point as a location can always only accomodate at the same time
one point of a material body or unpenetrateable object or being empty
or void.
Now i can analytically describe them by describing the space they
occupy at a certain point of time.
Every such object i can analyze into the union of open sets and of
points.
I never saw a definition for a point as a set, still we can take them
as geometrical elements, just like Euclid did in his ,,Elements". Bob
wrote:
> A point is not a set of things ...
And the open sets are line segments without their endpoints, areas
of surfaces without a rim (border-line) and bodies of volume without
their surfaces.
Two open bodies of volume can never touch ( and remember, they can not
penetrate), so when i want to ,,glue" them together i need to add at
least one point to ,,glue" them into one object.
But the point, piece of line or surface-area, the ,,glue", just lies
adjascent to the other points of the objects, i have to heat it up or
need a second component of glue. Otherwise a deformation can seperate
them.

Now a line is more then just a set of points, i learned that in
between two points is always an open intervall. And all points
collected into one set, and all open intervals collected into a
second set, together form the topology of a line ( with three rules
added). In a similar way i can build the topological plane and the
topological 3D-space.
An unpenetrateable object is not only a subset of the points of space,
it must be equipped with a set, containing all the open sets of the
topology of the space, which exists for these points. I think, this
is called subspace topology.

When i add to two open objects very close to each other at a certain
point of time just a point ( or segment of line or area) i have to add
not only their sets of points, and not only their sets of open sets, i
must provide more glue in form of some more sets of open sets.
The other way. When i remove a point from an intervall, it becomes two
intervals. And automatically their topolgy or set of subsets are
reduced.
Now - finally i can express, that what i think possible. When i remove
from a line segment not one point, but from the topology of it all
sets, containing one and the same point - then i have done a ,,cut" as
well. The result is two pieces of line, just lying next to each other,
no point removed and looking still unchanged. With the deformations of
topology these parts can move to seperated parts of space, but
deformation of the original piece will leave it always be one piece.
Did i do a topological proper cut? Or where is it miss? Or are there
other cut and paste operations?

With friendly greetings
Hero

.

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