Is it possible to construct a connected subset of the plane with the property that removal of any single point makes it totally disconnected? Any answer is appreciated..Thanks!!
2026-03-31 15:12:09.1774969929
A very weird connected subset of $\mathbb R^2$
392 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
No, it is impossible, if we exclude the trivial case of a singleton. Such a point is called a dispersion point. A space can only have one dispersion point. So we cannot remove any single point to obtain a totally disconnected space.
However, if we only want that the removal of one single point makes the space totally disconnected, then the Knaster-Kuratowski fan, or Cantor's teepee is a well-known example.