Proof That Rn With One Point Removed is Connected

Theorem

Ifisa point inthenisconnected.

Proof

Suppose P and P' are any two points ofAnyclosed line segment inishomeomorphic to the closed intervalhenceis a connected subspace ofChooseP'' in suchthatisnot on either of the lines PP'' or P'P'.

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

Henceisconnected.

P and P' belong to the connected subspace ofHenceisconnected.

Notice thatisnot connected with one point removed.