[Math] Quotient space is connected…

Question

Let $X$ be a topological space and $\sim$ be an equivalence relation defined on it. Let $Y$ be the space $X/{\sim}$ and $p :X \rightarrow Y $ be the quotient map, and give $Y$ the quotient topology. If $Y$ is connected, must $X$ be connected as well?

Answer 1

Not necessarily. Let $X$ be a non-connected space and let $\sim$ be the equivalence such that $X$ itself is the only equivalence class (equivalently $p$ is constant). Then $Y$ is a singleton, hence connected, but $X$ is not.