For some reason I am having some trouble on this basic point set topology question:
Suppose $X$ is connected, and $A$ is a connected subset of $X$, and that $B$ is a clopen set in $X-A$ (not in $X$, obviously. Clopen in the subspace topology on $X-A$).
Show $A \cup B$ is connected.
I'm getting a little confused on the correct way to approach this...i.e. which definition or notion of connectedness I should use to approach it.
Try to prove the following lemma: Let $X$ be a topological space, $X = A ∪ B$, $C ⊆ A ∩ B$ is clopen in both $A$ and $B$. Then $C$ is clopen in $X$.
Then take any $C$ clopen in your $A ∪ B$ and show that it is trivial.