= Solution
Give the nonempty <subset> $A\subseteq X$ its <subspace topology>. It is <compact> if every <open cover> admits a <finite subcover>. Equivalently, whenever $A\subseteq\bigcup_{j\in J}U_j$ with each $U_j$ <open> in $X$, there are $j_1,\ldots,j_m$ such that $A\subseteq U_{j_1}\cup\cdots\cup U_{j_m}$.
It is <connected> if it has no <separation of a topological space>: there are no disjoint nonempty <open subsets> $B,C$ of $A$ with $A=B\cup C$. Here openness is relative to the <subspace topology>; equivalently, the only <clopen sets> of $A$ are $\varnothing$ and $A$.
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