With Euclidean induced metrics, let $X=\{(0,0)\}\cup\{(x,\sin(1/x)):x\ne0\}$ and $Y=\{(0,0)\}\cup\{(x,x\sin(1/x)):x\ne0\}$. Let Bₓ,Bᵧ be their open unit balls about the origin. I: the closure of Bₓ in X is compact. II: the closure of Bᵧ in Y is compact.
For X, choose points tending to a missing limit (0,c) with 0 < |c| < 1 along oscillations of sin(1/x). They lie in the unit ball eventually but have no convergent subsequence in X, so I is false. The graph defining Y is continuous at zero because $|x\sin(1/x)|\leq|x|$. It is closed in R²; the closure of its unit ball is a closed bounded subset of that graph and hence compact. Thus II is true, giving C.
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