\magnification 2200 \parindent 0pt \parskip 12pt \hsize 7.2truein \vsize 9.7 truein \hoffset -0.45truein \def\w{\vskip -2pt} \def\u{\vskip -7pt} \def\v{\vskip -6pt} \def\h{\hskip 10pt} Topologies on $Y^X = \{f: X \rightarrow Y\}$ ${\cal T}_P$ = Product topology \v ${\cal T}_U$ = Uniform topology \v ${\cal T}_B$ = Box topology ${\cal T}_P$ = Point-open topology = topology of pointwise convergence: \v \h Let $S(x, U) = \{f ~|~ f \in Y^X, f(x) \subset U\}$ \v \h Subbasis = $\{ S(x, U) ~|~ x \in X, ~U~ open ~in ~Y\}$ \v ${\cal T}_C$ = Topology of compact convergence = topology of uniform convergence on compact sets: \v \h Let $B_C(f, \epsilon) = \{ g ~|~ sup\{d(f(x), g(x)) ~|~ x \in C\} < \epsilon \}$ \v \h Basis = $\{ B_C(f, \epsilon) ~|~ C~ compact, ~f \in Y^X, ~\epsilon > 0\}$ ${\cal T}_{CO}$ = Compact-open topology: \v \h $S(C, U) = \{ g ~|~ g \in C(X, Y), f(C) \subset U\}$ \v \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 $f|_C$. \w Defn: $X$ is compactly generated if \v \vskip -4pt $A$ open in $X$ if $A \cap C$ open in $C$ for each compact subspace $C$ of $X$. \v \vskip -3pt Or equivalently, \vskip -3pt \v $B$ is closed in $X$ if $B \cap C$ is closed in $C$ for each compact subspace $C$ of $X$. \w Lemma: 46.3\hfil \break $X$ locally compact implies $X$ compactly generated. \u\u $X$ first countable implies $X$ compactly generated. \w 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. \w 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. \w 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 ${\cal T}_P \subset {\cal T}_C \subset {\cal T}_U \subset {\cal T}_B$ \u Thm 46.8: If $Y$ is a metric space, compact open topology = topology of compact convergence on $C(X, Y)$. \u Cor: 46.9: If $Y$ is a metric space, the compact convergence topology on $C(X, Y)$ does not depend on the metric of $Y$. \u 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. \v 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 $F_Z: X \rightarrow Y$, $F_z(X) = f(x, z)$ \w\w\w\v $F$ is the map of $Z$ into $C(X, Y)$ induced by $f$. \v Conversely, given $F: Z \rightarrow C(X, Y)$, define $f: X \times Z \rightarrow Y$ by $f(x, z) = (F(z))(x)$ \u 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. \end