Let $X, Y$ be topological spaces and let $\text{Cont}(X,Y)$ be the collection of continuous functions $f:X\to Y$. We say that a topology $\tau$ on $\text{Cont}(X,Y)$ is admissible if the evaluation map $$e: \text{Cont}(X,Y)\times X\to Y; \ (f,x)\mapsto f(x)$$ is continuous. Clearly the intersection of all admissible topologies on $\text{Cont}(X,Y)$, which we call $\tau_{\text{Cont}}$ is admissible.

Questions:

1. Is there a name for $\tau_{\text{Cont}}$?

2. What is an example of spaces where $\tau_{\text{Cont}}$ is not the compact-open topology?

Let $X, Y$ be topological spaces and let $\text{Cont}(X,Y)$ be the collection of continuous functions $f:X\to Y$. We say that a topology $\tau$ on $\text{Cont}(X,Y)$ is admissible if the evaluation map $$e: \text{Cont}(X,Y)\times X\to Y; \ (f,x)\mapsto f(x)$$ is continuous.

Is there an example of spaces $X,Y$ such that the intersection of all admissible topologies on $\text{Cont}(X,Y)$ is no longer admissible?