I'm having trouble understanding how deleting a point from R2 leaves it a connected space, while deleting a point from R doesn't do so.
Can someone help me with this?
Thanks,
MAX
This is one characterization of connect sets:
A set is connected if it is not the union of two non-empty separated sets.
Two sets are separated if neither contains a point or a limit point of the other.
Now $\displaystyle \Re \backslash \{ a\} = \left( { - \infty ,a} \right) \cup \left( {a,\infty } \right)$, clearly the union of two separated sets.
Try that with $\displaystyle \Re \times \Re \backslash \{ (a,b)\} $. What two separated sets would be possible?
There are several different notations.
$\displaystyle \Re ^2 = \Re \times \Re = \left\{ {\left( {x,y} \right):x \in \Re \wedge y \in \Re } \right\}$
So $\displaystyle \Re ^2 \backslash \left\{ {(a,b)} \right\} = \left\{ {\left( {x,y} \right) \in \Re ^2 :x \ne a \wedge y \ne b} \right\}$