Suppose $B_{\epsilon}$ are closed subsets of a compact space and $B_{\epsilon} \supset B_{\epsilon'} \quad \forall \epsilon > \epsilon'$. Furthermore, $B_0 = \bigcap_{\epsilon>0} B_{\epsilon}$. For a continuous function $f$ can we conclude that $$f(B_0) = \bigcap_{\epsilon>0} f(B_{\epsilon})?$$
I believe the answer to be yes. It seems this should be a well-known property---I'm having trouble finding a reference.