Could someone explain why what's in the red box is true? How does the compactness of $\Phi_e^{-1}(C \cap \bar e)$ imply the compactness of $C \cap \bar e$?
2026-03-27 23:14:04.1774653244
If $\Phi_e^{-1}(C \cap \bar e)$ is compact, then $C \cap \bar e$ is compact?
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Since $\Phi_e$ is surjective onto $\overline{e}$, $\Phi_e(\Phi_e^{-1}(C\cap \overline{e}))=C\cap\overline{e}$. Thus $C\cap \overline{e}$ is the continuous image of a compact set and hence compact.