159 lines
4.0 KiB
Markdown
159 lines
4.0 KiB
Markdown
# Math4201 Lecture 16 (Topology I)
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## Continuous maps
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The following maps are continuous:
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$$
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F_+:\mathbb{R}\times \mathbb{R}\to \mathbb{R}, (x,y)\to x+y
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$$
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$$
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F_-:\mathbb{R}\times \mathbb{R}\to \mathbb{R}, (x,y)\to x-y
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$$
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$$
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F_\times:\mathbb{R}\times \mathbb{R}\to \mathbb{R}, (x,y)\to x\times y
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$$
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$$
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F_\div:\mathbb{R}\times (\mathbb{R}\setminus \{0\})\to \mathbb{R}, (x,y)\to \frac{x}{y}
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$$
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### Composition of continuous functions is continuous
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Let $f,g:X\to \mathbb{R}$ be continuous functions. $X$ is topological space.
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Then the following functions are continuous:
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$$
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H:X\to \mathbb{R}\times \mathbb{R}, x\to (f(x),g(x))
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$$
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Since the composition of continuous functions is continuous, we have
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$$
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F_+\circ H:X\to \mathbb{R}, x\to f(x)+g(x)
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$$
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$$
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F_-\circ H:X\to \mathbb{R}, x\to f(x)-g(x)
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$$
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$$
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F_\times\circ H:X\to \mathbb{R}, x\to f(x)\times g(x)
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$$
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are all continuous.
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More over, if $g(x)\neq 0$ for all $x\in X$, then
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$$
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F_\div\circ H:X\to \mathbb{R}, x\to \frac{f(x)}{g(x)}
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$$
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is continuous following the similar argument.
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### Defining metric for functions
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#### Definition of bounded metric space
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A metric space $(Y,d)$ is **bounded** if there is $M\in\mathbb{R}^{\geq 0}$ such that
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$$
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\forall y,y'\in Y, d(y,y')<M
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$$
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<details>
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<summary>Example of bounded metric space</summary>
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If $(Y,d)$ is a bounded metric space, let $M$ be a positive constant, then $\overline{d}=\min\{M,d\}$ is a bounded metric space.
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In fact, the metric topology by $d$ and $\overline{d}$ are the same. (proved in homeworks)
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</details>
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Let $X$ be a topological space. and $(Y,d)$ be a **bounded** metric space.
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$$
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\operatorname{Map}(X,Y)\coloneq \{f:X\to Y|f \text{ is a map}\}
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$$
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Define $\rho:\operatorname{Map}(X,Y)\times \operatorname{Map}(X,Y)\to \mathbb{R}$ by
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$$
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\rho(f,g)=\sup_{x\in X} d(f(x),g(x))
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$$
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#### Lemma space of map with metric defined is a metric space
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$(\operatorname{Map}(X,Y),\rho)$ is a metric space.
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<details>
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<summary>Proof</summary>
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Proof is similar to showing that the square metric is a metric on $\mathbb{R}^n$.
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$\rho(f,g)=0\implies \sup_{x\in X}(d(f(x),g(x)))=0$
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Since $d(f(x),g(x))\geq 0$, this implies that $d(f(x),g(x))=0$ for all $x\in X$.
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The triangle inequality of being metric for $\rho$ follows from the similar properties for $d$.
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</details>
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#### Lemma continuous maps form a closed subset of the space of maps
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Let $(\operatorname{Map}(X,Y),\rho)$ be a metric space defined before.
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and
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$$
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Z=\{f:X\to Y|f \text{ is a continuous map}\}
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$$
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Then $Z$ is a closed subset of $(\operatorname{Map}(X,Y),\rho)$.
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<details>
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<summary>Proof</summary>
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We need to show that $\overline{Z}=Z$.
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Since $\operatorname{Map}(X,Y)$ is a metric space, this is equivalent to showing that: Let $f_n:X\to Y\in Z$ be a sequence of continuous maps,
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Which is to prove the uniform convergence,
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$$
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f_n \to f \in \operatorname{Map}(X,Y)
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$$
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Then we want to show that $f$ is also continuous.
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It is to show that for any open subspace $V$ of $Y$, $f^{-1}(V)$ is open in $X$.
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Take $x_0\in f^{-1}(V)$, we'd like to show that there is an open neighborhood $U$ of $x_0$ such that $U\subseteq f^{-1}(V)$.
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Since $x_0\in f^{-1}(V)$, then $f(x_0)\in V$. By metric definition, there is $r>0$ such that $B_r(f(x_0))\subseteq V$.
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Take $N$ to be large enough such $\rho(f_N(x), f(x)) < \frac{r}{3}$
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So $\forall x\in X$, $d(f(x),f_N(x))<\frac{r}{3}$
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Since $f_N$ is continuous, $f_N^{-1}(B_{r/3}(f(x_0)))$ is an open set $U\subseteq X$ containing $x_0$.
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Take $x\in U$, $d(f(x),f(x_0))<d(f(x),f_N(x_0))+d(f_N(x),f_N(x_0))+d(f_N(x_0),f(x_0))$ using triangle inequality.
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Note that,
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$d(f(x),f_N(x))<\frac{r}{3}$ (using $N$ large enough),
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$d(f_N(x),f_N(x_0))<\frac{r}{3}$ (using $x\in U$, then $f_N(x)\in B_{r/3}(f_N(x_0))$, so $d(f_N(x),f_N(x_0))<\frac{r}{3}$),
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$d(f_N(x_0),f(x_0))<\frac{r}{3}$ (using $N$ large enough),
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So $d(f(x),f(x_0))<\frac{r}{3}+\frac{r}{3}+\frac{r}{3}=r$.
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So $f(x)\in B_r(f(x_0))\implies x\in f^{-1}(B_r(f(x_0)))\implies x\in f^{-1}(V)\implies U\subseteq f^{-1}(V)$.
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So $f^{-1}(V)$ is open in $X$.
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</details> |