User:Lethe/evaluation

problem
Show that the evaluation map e: X×Top(X,Y) → Y given by (x,f) ↦ f(x) is continuous if X is a locally compact Hausdorff space (this can apparently be weakened to regularity) and Top(X,Y) (the hom-set between X and Y, that is, the continuous maps between X and Y) carries the compact open topology.

proof
Let V be an open set in Y, and (x0,f0) be an element of e−1(V). We want so show that e−1(V) is open, i.e. that every pair (x0,f0) is contained in a neighborhood in e−1(V) open in Top(X,Y).

Since X is locally compact, every open neighborhood of x0 contains a compact subset. f0(x0) ∈ V, therefore U=f0−1(V) contains x0. It is open since f0 is continuous. Let K be a compact neighborhood of x0 in U, and M an open neighborhood of x0 in K. K⊆U, so f0(K) ⊆ V, and so f0∈W(K,V), the set of continuous maps φ∈Top(X,Y) such that φ(K)⊆V, which is open in the compact-open topology by definition.

Thus (x0,f0)∈M×W(K,V), and it remains to show that M×W(K,V) is contained in e−1(V). For any (ξ,φ)∈ M×W(K,V), since ξ∈K, then we have e(ξ,φ)=φ(ξ)∈V since φ(K)⊆V.

Therefore every point in M×W(K,V) is also in e−1(V), i.e. M×W(K,V)⊆e−1(V). Since every point (x0,f0) of e−1(V) has an open neighborhood contained in e−1(V), e−1(V) is open, and e is continuous.

Somehow, I don't seem to have used that X is Hausdorff anywhere in this proof. So either my proof is wrong, or I've used it somewhere implicitly without knowing it.