Let S Connected subset of R^2 and p $\displaystyle \in$ R^2 accumulation point. Show that {p} union S is conected. What happends if p isnt a acummulation point?
Proofs on connected sets depend on the sequence of text material you have.
Look for a theorem about the existence of two disjoint open sets that separate $\displaystyle \{ p \}\cup S$.
Note any open set that contains $\displaystyle p$ must contain another point of $\displaystyle S$.
But would that separate $\displaystyle S$?