Pointwise topology embedding

I'll show continuity. The pointwise topology on $C(X,Y)$ is precisely what it says: a net $g_i$ converges in it, $g_i\to g$, if and only if for all $x\in X$ we have $g_i(x)\to g(x)$ in $Y$. (Indeed, this is because a net in a product converges iff all projections converge.)

So suppose $f_i\to f$ in $C(Z\times X,Y)$. We need to show $\Lambda (f_i)\to \Lambda (f)$. The latter means that, for all $z\in Z$, we have $(f_i)_z\to f_z$ in $C(X,Y)$ . But this means that for all $z\in Z$, we have that for all $x\in X$, $(f_i)_z(x)\to f_z(x)$ in $Y$. In other words, it means that for all $(z,x)\in Z\times X$, we have $f_i(z,x)\to f(z,x)$ in $Y$. But this is precisely our assumption $f_i\to f$ in $C(Z\times X,Y)$.

So it's basically a tautology: the maps are defined by 'currying' (holding the function pointwise fixed), and we are considering the pointwise topology.

To show that in fact $\Lambda(f)$ is continuous for continuous $f$, a similar argument can be used (namely $\Lambda(f)(z_i)\to \Lambda(f)(z)$ for nets $z_i\to z$), and the same for the well-definedness and continuity of the inverse.

(I think this is one of the places where using nets is superior to working directly with open set / subbases of the topology.)