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If $E$ is a Banach space, denote with $I$ the identity map of $E$, denote the space of continuous functions from $E$ to $E$ with $\operatorname{C}(E,E)$ and denote the space of homeomorphism from $E$ onto itself with $\operatorname{Homeo}(E)$.

Definition 1: If $E$ is a Banach space, $\emptyset\neq A\subset K\subset E$ and $O\subset E$, we say that $O$ strongly links $(K,A)$ if $$\forall\varphi\in \operatorname{C}(E,E), (\varphi|_A=I|_A)\implies (\varphi(K)\cap O\neq \emptyset).$$

And

Definition 2: If $E$ is a Banach space, $\emptyset\neq A\subset K\subset E$ and $O\subset E$, we say that $O$ weakly links $(K,A)$ if $$\forall\varphi\in \operatorname{Homeo}(E,E), (\varphi|_A=I|_A)\implies (\varphi(K)\cap O\neq \emptyset).$$

Clearly, if $O$ strongly links $(K,A)$ then $O$ weakly links $(K,A)$.

Main question: If $O$ weakly links $(K,A)$, is it also true that $O$ strongly links $(K,A)$?

A secondary question: if the answer to the main question is no, are there any sensible topological properties (e.g. compactness, connectedness, being the closure of an open set, being $A$ the boundary of $K$...) that we can impose on $A$ or $K$ or $O$ in order to get the equivalence of the two definitions?

Comments: In order to avoid pathologies, I set the question in the context of Banach spaces, where I need it in the context of generalization of theorems of mountain pass type.

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