Proposition 1

— Let X and Y be two topological spaces and f a continuous mapping of X into Y. Let $(K_\alpha)_{\alpha \in A}$ be a decreasing directed family of compact subsets of X, with intersection K. Then $f(K) = \bigcap_{\alpha \in A} f(K_\alpha)$.

For, let y be a point of $\bigcap_{\alpha \in A} f(K_\alpha)$; for every $\alpha \in A$, the set $L_\alpha = K_\alpha \cap f^{-1}(y)$ is compact and nonempty. The family $(L_\alpha)_{\alpha \in A}$ is directed downward, therefore its intersection L is nonempty. Now, $L = K \cap f^{-1}(y)$, whence $y \in f(K)$. We have thus proved the inclusion $f(K) \supset \bigcap_{\alpha \in A} f(K_\alpha)$, and the reverse inclusion is obvious.