I am trying to complete this topological question and I would like to know if my solution is correct. Any help would be greatly appreciated!
My Solution:
Let X be a topological space, and Y a Hausdorff topological space. Now, let f,g $\in$ Hom(X,Y) be such that f $\neq$ g. Then, there $\exists$ $x$$_o$$\in$ $x$ such that f($x$$_o$)$\neq$ g($x$$_o$). Since Y is Hausdorff, there $\exists$ open neighborhoods U of f($x$$_o$) and V of g($x$$_o$) such that U$\cap$V=$\emptyset$. Then, f $\in$ S({$x$$_o$},U) and g $\in$ S({$x$$_o$},V). Also, S({$x$$_o$},U) $\cap$ S({$x$$_o$},V) = $\emptyset$, becuase if h $\in$ S({$x$$_o$},U) $\cap$ S({$x$$_o$},V), then h($x$$_o$) $\in$ U and h($x$$_o$) $\in$ V. Since, U $\cap$ V = $\emptyset$, this implies that h maps $x$$_o$ into two distinct points. Thus we have a contradiction. Therefore, Hom(X,Y) is Hausdorff with the compact open topology.
Your proof is absolutely correct.