Topology on function spaces

Ways to motivate:
- we want a notion of when $f_k \to f$
  ...but there's several notions: which is -good-?
 
- more globally, we want a topology on $C(X,Y)$
  (IE, continuous functions $X \to Y$)
  s.t. the evaluation map
  $C(X,Y) \times X \to Y$
  by
  $f,x \mapsto f(x)$
  is continuous
IE, if $f_k \to f$ and $x_k \to x$,
then $f_k(x_k) \to f(x)$
  note the diagonalization-type argument:
  a sequence in $C(X,Y) \times X$
  is a sequence $(f_k,x_k)$.
the uniform topology (generally, compact-open)
is the -weakest- topology in which this works,
I think...

- also, we want $C(X,Y)$ to be closed in $F(X,Y)$
  indeed, since we have a metric structure (well, on compacts)
  we want it to be -complete- (and locally compact??)

- pointwise is too coarse:
  tents example shows $C(X,Y) \times X \to Y$ not continuous
  (we have a seq. of -continuous- functions
   converging (pointwise) to a continuous function
   s.t. $f_k(x_k)$ doesn't go to $f(x)$:

  $x^n$ example shows $C(X,Y)$ not closed in $F(X,Y)$

  NB: see how we're giving -abstract- explanations of what each
  example -shows-.
  Concretely, of course, what we're saying is:
  pointwise is bad b/c:
  - limit of cont. needn't be continuous;
  - even when converge to continuous, pointwise conv.
    doesn't commute with conv. of seq. of points

- $1$-parameter family of functions:
  here's another point:
  we want to clarify what we mean by a
  ``$1$-parameter family of functions''
  with means we want to know what a continuous function
  $I \to F(X,Y)$ means
  ...but turns out adjunction is usually easier...

  NB: actually, we also want to talk about ``smoothly varying
  families'', and putting a differentiably structure on
  infinite-dimensional spaces is too damn hard for kids
  (and perhaps for adults???)
  ...so we answer it using adjunction!
  IE, the topology on a function space should agree with:
  $C(X \times Y, Z) = C(X,C(Y,Z))$

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What's wrong with the pointwise topology? (on $F(X,Y)$?)
It ignores the topology on $X$!
That is, if $X$ is discrete, then the pointwise topology is the
correct one to put on $F(X,Y)$.

The uniform topology kinda deals with it, but kinda says
that you're putting the -trivial- topology on $X$ (not quite, b/c
you're talking about a uniform structure on $Y$, but kinda):
something has to be true for -all- points at once.

The compact-open topology is better, and actually cares about the
topology on $X$ (which none of the previous topologies do).
