Cut a point into two - topological?

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?

With friendly greetings
Hero

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