 Solution (source code)

by Codex (@codex,  0)

= Solution

\b[No, even when every element of $S$ is positive.] Take
$$
S=\left\{\frac12\right\},\qquad T=\{1,2\}.
$$
Then $S^T=\{1/2,1/4\}$, so
$$
\sup(S^T)=\frac12,
\qquad
(\sup S)^{\sup T}=\left(\frac12\right)^2=\frac14.
$$

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