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Topologies on $Y^X = \{f:  X \rightarrow Y\}$

${\cal T}_P$ = 
Product topology
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${\cal T}_U$ = 
Uniform topology
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${\cal T}_B$ = 
Box topology

${\cal T}_P$ = 
Point-open topology  = topology of pointwise convergence:
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\h   Let  $S(x, U) = \{f ~|~ f \in Y^X, f(x) \subset U\}$
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\h   Subbasis = $\{ S(x, U)  ~|~ x \in X, ~U~ open ~in ~Y\}$
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${\cal T}_C$ = 
Topology of compact convergence = topology of uniform convergence on compact sets:
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\h   Let $B_C(f, \epsilon) = \{ g ~|~ sup\{d(f(x), g(x)) ~|~ x \in C\} < \epsilon \}$
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\h    Basis = $\{ B_C(f, \epsilon)  ~|~ C~ compact, ~f \in Y^X, ~\epsilon > 0\}$


${\cal T}_{CO}$ = 
Compact-open topology:  
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\h   $S(C, U) = \{ g ~|~ g \in C(X, Y), f(C) \subset U\}$
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\h    Subbasis = $\{ S(C, U) ~|~ C ~ compact~ in ~ X, ~U~ open~ in~ Y \}$

Thm 46.1:  $f_n$ converges to $f$ in the point-open topology if and only if for all $x \in X$, $f_n(x)$ converges to $f(x)$ in $Y$.

Thm 46.2:  $f_n$ converges to $f$ in the topology of compact convergence if and only if for each 
compact subspace $C$ of $X$, $f_n|_C$ converges uniformly to $f|_C$.
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Defn:  $X$ is compactly generated if
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$A$ open in $X$ if $A \cap C$ open in $C$  for each compact subspace $C$ of $X$.
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Or equivalently,
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$B$ is closed in $X$ if $B \cap C$ is closed in $C$  for each compact subspace $C$ of $X$.
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Lemma: 46.3\hfil \break
  $X$ locally compact implies $X$ compactly generated.
\u\u
$X$ first countable implies $X$ compactly generated.
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Lemma 46.4: If $X$ is compactly generated, then $f: X \rightarrow Y$ is continuous of for each compact 
$C$ of $X$, $f|_C$ is continuous.  
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Cor 46.6:  If $X$ is compactly generated and $Y$ is a metric space, then if $f_n$ is continuous and 
$f_n $ converges to $f$ in the topology of compact convergence, then $f$ is continuous.
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Cor 46.5: If $X$ is compactly generated and $Y$ is a metric space, then $C(X, Y)$ is closed in $Y^X$ 
in the 
topology of compact convergence.



Thm 46.7: If $Y$ is a metric space
 ${\cal T}_P \subset {\cal T}_C \subset {\cal T}_U \subset {\cal T}_B$
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Thm 46.8:  If $Y$ is a metric space, compact open topology = topology of compact 
convergence on $C(X, Y)$.
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Cor:  46.9:  If $Y$ is a metric space, the compact convergence topology on $C(X, Y)$ does not depend 
on the metric of $Y$.
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Cor:  If $X$ compact, the uniform topology on $C(X, Y)$ does not depend on the metric of $Y$.

Defn:  $e: X \times C(X, Y) \rightarrow Y$, $e(x, f) = f(x)$ is called the evaluation map.
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Thm 46.10:  If $X$ is locally compact Hausdorff and $C(X, Y)$ has the compact open topology, then $e$ 
is continuous.


Defn:  Given $f:X \times Z \rightarrow Y$, define $F: Z \rightarrow C(X, Y)$ by 
$F(z) = F_z$ where $F_Z: X \rightarrow Y$, $F_z(X) = f(x, z)$
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$F$ is the map of $Z$ into $C(X, Y)$ induced by $f$.
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Conversely, given $F: Z \rightarrow C(X, Y)$, define $f: X \times Z \rightarrow Y$
 by $f(x, z) = (F(z))(x)$
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Thm 46.11:  If $C(X, Y)$ has the compact-open topology, then $f$ continuous implies the induced 
function $F$ is 
continuous.  The converse holds if $X$ is locally compact Hausdorff.

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