0.1  (§46) Pointwise and compact convergence

• Notation. $Y^X=\{f:X\to Y\}={\prod }_{X}Y$.

• Def. For $x\in X$, $U\subset Y$ open:   $S(x,U)=\{f\mid f(x)\in U\}={\pi }_x^{-1}(U)$. These form a subbasis for the topology of pointwise convergence $=$ the product topology on $Y^X$.

• Lem. $f_n\to f$ in it $\iff$ $f_n(x)\to f(x)$ for every $x$.

• Def. $(Y,d)$ metric.   is a basis for the topology of uniform convergence.

  Proof. Basis axiom: for $g\in B(f,\epsilon )$ take $\delta =\epsilon -{sup}_{x}d(f,g)$; then $B(g,\delta )\subset B(f,\epsilon )$.

• Lem. $f_n\to f$ in it $\iff$ $f_n\to f$ uniformly.

• $\mathrm{★}$ Thm. A uniform limit of continuous functions is continuous.

• Def. For $C\subset X$ compact:   is a basis for the topology of compact convergence (uniform convergence on compact sets).

• Lem. $f_n\to f$ in it $\iff$ $f_n|C\to f|C$ uniformly for every compact $C\subset X$.

• Rmk. Ordering: pointwise $\subset$ compact $\subset$ uniform. If $X$ is compact the last two agree.

• Ex. $f_n(x)={\sum }_{k=0}^n{x}^k/k!\to e^x$ on $ℝ$: pointwise and uniformly on compact sets, but not uniformly.

• Def. $X$ is compactly generated if: $A\subset X$ is open whenever $A\cap C$ is open in $C$ for every compact $C\subset X$.

• Lem. Equivalently, with “closed” in place of “open”.

• Lem. $X$ compactly generated $\Rightarrow$ ($f:X\to Y$ continuous $\iff$ every $f|C$ is).

  Proof. $f^{-1}(U)\cap C={(f|C)}^{-1}(U)$.

• $\mathrm{★}$ Thm. $X$ compactly generated, $(Y,d)$ metric. If continuous $f_n\to f$ uniformly on compact subspaces, then $f$ is continuous.

• Prop. Locally compact spaces, and first-countable (hence metrizable) spaces, are compactly generated. [Mk p. 284]

0.2  Mapping spaces and the compact-open topology

• Notation. $𝒞(X,Y)=\{f:X\to Y\mid f\text{ continuous}\}\subset Y^X$. (Also written $\mathrm{Map}(X,Y)$.)

• $\mathrm{★}$ Def. For $C\subset X$ compact and $U\subset Y$ open: $S(C,U)=\{f\in 𝒞(X,Y)\mid f(C)\subset U\}.$ These form a subbasis for the compact-open topology.

• Rmk. Finer than pointwise convergence, since points are compact.

• $\mathrm{★}$ Thm. $X$ a space, $(Y,d)$ metric. On $𝒞(X,Y)$ the compact-open topology equals the topology of compact convergence.

  Proof. Two inclusions. $(\supset )$ Given $f\in S(C,U)$: $f(C)$ is compact in the open $U$, so $B(f(C),\epsilon )\subset U$ for some $\epsilon >0$; then $B_C(f,\epsilon )\subset S(C,U)$. $(\subset )$ Given $B_C(f,\epsilon )$: cover $C$ by finitely many $V_{x_i}$ with $f({\overline{V}}_{x_i})$ of diameter ; set $C_i={\overline{V}}_{x_i}\cap C$, $U_i=B(f(x_i),\epsilon /3)$; then $f\in {\bigcap }_{i}S(C_i,U_i)\subset B_C(f,\epsilon )$.

• Cor. Hence the compact-convergence topology on $𝒞(X,Y)$ depends only on the topology of $Y$, not on the chosen metric.

0.3  Joint continuity and the exponential law

• Def. $F:X\times Y\to Z$ and $f:X\to Z^Y$ correspond by $f(x)(y)=F(x,y).$ Call $F$ the left adjoint, $f$ the right adjoint.

• $\mathrm{★}$ Thm. $F$ continuous $\Rightarrow$ $f:X\to 𝒞(Y,Z)$ is continuous (compact-open topology).

  Proof. Given $f(p)\in S(C,U)$, i.e. $F(\{p\}\times C)\subset U$: apply the tube lemma to the open $F^{-1}(U)\supset \{p\}\times C$ with $C$ compact, getting $V\ni p$ with $V\times C\subset F^{-1}(U)$.

• $\mathrm{★}$ Prop. $Y$ locally compact Hausdorff $\Rightarrow$ the evaluation map $e:𝒞(Y,Z)\times Y\to Z,e(g,y)=g(y)$ is continuous.

  Proof. Given $g(p)\in U$: local compactness gives $V\ni p$ with $\overline{V}$ compact, $\overline{V}\subset g^{-1}(U)$. Then $W=S(\overline{V},U)\times V$ works.

• Thm. $Y$ locally compact Hausdorff: $f:X\to 𝒞(Y,Z)$ continuous $\Rightarrow$ $F:X\times Y\to Z$ continuous.

  Proof. $F=e\circ (f\times \mathrm{id})$.

• $\mathrm{★}$ Cor. $I=[0,1]$ is compact Hausdorff, hence locally compact Hausdorff. So there are bijective correspondences between maps

  $$F:X\times I\to Y,f:X\to 𝒞(I,Y),G:I\times X\to Y,g:I\to 𝒞(X,Y)$$

  ($g$ requires $X$ locally compact Hausdorff), related by $F(x,t)=f(x)(t)=G(t,x)=g(t)(x).$

• Rmk. Each such map is a homotopy between $g(0)$ and $g(1)$. This is the bridge into Chapter 7: a homotopy is a path in a mapping space.