= Solution
\b[True.] The <canonical projection of a quotient set> $p:X\to X/{\sim}$ is a <continuous> <surjection>. Its image is <compact> by the <continuous image of a compact space> theorem.
For <connectedness>, if $U,V$ formed a <separation of a topological space> of $X/{\sim}$, then the <preimages> $p^{-1}(U),p^{-1}(V)$ would be disjoint nonempty <open sets> covering $X$, contradicting its <connectedness>. This proves the <continuous image of a connected space> theorem in this case. Hence the <quotient space> is both <compact> and <connected>.
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