= Solution
\b[False.] Removing a point from the <real line> leaves two disjoint nonempty open rays, so the remainder is disconnected. Removing a point from the <Euclidean plane> leaves a <path-connected space>: join any two remaining points by a <line segment> if it avoids the removed point, or by two segments through a point not on either of the two lines through the removed point and the chosen endpoints.
A proposed <homeomorphism> would restrict to a <homeomorphism> between these punctured spaces and preserve <connectedness>, which is impossible. Hence
$$
\boxed{\mathbb R^2\not\cong\mathbb R.}
$$
Back to article page
