#hemicontinuous

The equalizer of f and g is defined as: Where f,g : X→Y Eq(f,g) = {x∈X|f(x)=g(x)}

Let X be a locally compact Hausendorff, and Y some topological space. (reminder/explanation because I had to look these up myself: Hausendorff means for any 2 points in the space, there exists a neighborhood of each of the two points such that the intersection of the neighborhoods is empty. This is true for lots of spaces one would probably look at. Tbh I don’t exactly know what locally compact means, but I know Hausendorff and compact implies locally compact, and that compact means any open cover has a finite subcover. ) Let F be the space of continuous functions from X to Y, with the Compact-open topology. Let F×X have the product topology. Let f be some function in F (and therefore be continuous). Let f be ubiquitous, i.e. for any g in F, Eq(f,g) is not empty.

Define h:F→Powerset(X) as h=Eq(f,·). i.e. h(g)=Eq(f,g).

Claim: h is upper hemicontinuous.

Consider g_n a convergent sequence in F which converges to g_inf, and a convergent sequence x_n such that for each n, x_n ∈ g_n , and the x_n converge to x_inf .

For all n, x_n in h(g_n) , i.e. x_n in Eq(f,g_n) , i.e. f(x_n)=g_n(x_n) . So for all n, g_n(x_n)=f(x_n).

Because the g_n converge to g_inf in F , and the x_n converge to x_inf in X, the sequence (g_n,x_n) converges to (g_inf,x_inf) in F×X. It is a theorem that eval:C(X,Y)×X→Y with eval(g,x)=g(x) is continuous when C(X,Y) has the compact-open topology and X is locally compact Hausendorff, and C(X,Y)×X has the product topology. So, eval:F×X→Y is continuous. So, because (g_n,x_n) converges to (g_inf,x_inf), eval(g_n,x_n) converges to eval(g_inf,x_inf). eval(g_inf,x_inf)=g_inf(x_inf), and eval(g_n,x_n)=g_n(x_n).

From before, for all n, g_n(x_n)=f(x_n).

f(x_n) converges to f(x_inf) because x_n converges to x_inf, and f is continuous.

So, g_n(x_n) converges to g_inf(x_inf), but g_n(x_n)=f(x_n), and f(x_n) converges to f(x_inf). Therefore, g_inf(x_inf)=f(x_inf). So, x_inf is in Eq(f,g_inf). So, x_inf is in h(g_inf).

So, we have shown that for any g_n converging to g_inf in F, and any x_n in h(g_n) , with x_n converging to x_inf in X, x_inf is in h(g_n).

Therefore, h is upper hemicontinuous (as this is the definition of upper hemicontinuity).

But wait, none of this required that f be ubiquitous, so that isn’t actually required. f can be any function in F.

But I learned all this in order to try to understand things about ubiquitous functions, so I put it there. Also, it is nice that it being ubiquitous means that the output of h is never the empty set.

So, yeah, this proof, and this post, are done.

Oh, also, h(g) is always closed. If X is compact, then h(g) is compact.