I have come across a question asking to prove/disprove that $\mathbb Q^+$, the one point compactification of $\mathbb Q$ endowed with subspace topology inherited from $\mathbb R$, is connected.
I think $\mathbb Q^+$ is connected.
$\mathbf {My Attempt.}$ Claim: Let $A,B$ be two nonempty open set in $\mathbb Q^+$ such that $\mathbb Q^+=A\cup B$, then $A\cap B \neq \emptyset$
Let $A,B$ be two nonempty open set in $\mathbb Q^+$ such that $\mathbb Q^+=A\cup B$. Without loss of generality, let $\infty\in A$. Then according to the definition of topology on $\mathbb Q^+$, $\mathbb Q^+ -A$ is closed and compact in $\mathbb Q$. Since only compact spaces in $\mathbb Q$ endowed with subspace topology inherited from $\mathbb R$, is finite sets, thus $\mathbb Q^+ -A$ must be finite. Again any open set in $\mathbb Q^+$ is infinite. Hence $B$ is also infinite. Since $\mathbb Q^+ -A$ is finite, thus $B$ must intersect $A$. Hence $A\cap B\neq\emptyset$.
So $\mathbb Q^+$ cannot have a separation. Hence $\mathbb Q^+$ is connected.
Am I right? Please give your suggestions. Thank you.