Proof That Rn With One Point Removed is Connected

Theorem

Ifis a point inthenis connected.

Proof

Suppose P and P' are any two points ofAny closed line segment inis homeomorphic to the closed intervalhence is a connected subspace ofChoose P'' in such thatis not on either of the lines PP'' or P'P'.

The sets PP'' and P''{' are connected with nonempty intersection since P'' lies on both lines.

Henceis connected.

P and P' belong to the connected subspace ofHenceis connected.

Notice thatis not connected with one point removed.