From 21ff5779ecc49208833d75498aa3c4f0b828d201 Mon Sep 17 00:00:00 2001 From: Robert Date: Sat, 27 Mar 2021 15:49:41 +0100 Subject: [PATCH] Finished open and closed set section --- chapters/sections/open_closed_sets.tex | 338 +++++++++++++++++++++++++ chapters/topo_of_metr_spaces.tex | 1 + script.pdf | Bin 671742 -> 712812 bytes script.tex | 5 +- 4 files changed, 343 insertions(+), 1 deletion(-) create mode 100644 chapters/sections/open_closed_sets.tex diff --git a/chapters/sections/open_closed_sets.tex b/chapters/sections/open_closed_sets.tex new file mode 100644 index 0000000..da783d1 --- /dev/null +++ b/chapters/sections/open_closed_sets.tex @@ -0,0 +1,338 @@ +\documentclass[../../script.tex] {subfiles} +%! TEX root = ../../script.tex + +\begin{document} +\section{Open and Closed Sets} + +\begin{defi}[Inner points and Boundary points] + Let $\metric$ be a metric space, $A \subset X$ and $x \in X$. + \begin{enumerate}[(i)] + \item $x$ is said to be an inner point of $A$, if \[ \exists \epsilon > 0: ~~ \oball(x) \subset A \] + \item $x$ is said to be a boundary point of $A$ if + \[ + \forall \epsilon > 0: ~~\underbrace{\oball(x) \cap A \ne \emptyset}_{\substack{\oball(x) \text{ contains} \\ \text{points from } A}} + \wedge \underbrace{\oball(x) + \cap (X \setminus A) \ne \emptyset}_{\substack{\oball(x) \text{ contains points}\\ \text{from outside of } A}} + \] + \item The set + \[ + \set[x \text{ is inner point of } A]{x \in X} + \] + is called the interior of $A$, and is denoted as $\interior{A}$. + \item The set + \[ + \set[x \text{ is boundary point of} A]{x \in X} + \] + is called the boundary of $A$, and is denoted as $\partial A$. + \item $A \cup \boundary{A}$ is said to be the closure of $A$, and is denoted as $\closure{A}$. + \end{enumerate} + + \begin{center} + \begin{tikzpicture} + \draw [->, thick] (-6, -3.5) -- (-6, 3); + \draw [->, thick] (-6, -3.5) -- (2, -3.5); + \draw [thick, fill=lightgray] plot[smooth, tension=.7] coordinates {(-4,2.5) (-3,3) (-2,2.8) (-0.8,2.5) (-0.5,1.5) (0.5,0) (0,-2)(-1.5,-2.5) (-4,-2) (-3.5,-0.5) (-5,1) (-4,2.5)}; + + \node[below right] at (-6, 3) (X) {$X$}; + \node[below right] at (-4.5, 2) (A) {$A$}; + + \draw[fill] (-2, -1) circle [radius=1pt]; + \node[below right] at (-2, -1) (C) {$x$}; + \draw[dashed] (-2, -1) circle [radius=20pt]; + \node[below] at (-2, -1.75) (C) {$\oball(x)$}; + + \draw[fill] (-0.63, 2) circle [radius=1pt]; + \node[right] at (-0.63, 2) (C) {$x$}; + \end{tikzpicture} + \end{center} +\end{defi} + +\begin{eg} + Consider $X = \realn^2$. Then + \begin{align*} + A &= \set[0 \le y < 1]{(x, y) \in \realn} \\ + \mathring{A} &= \set[0 \le y < 1]{(x, y) \in \realn^2} \\ + \boundary{A} &= \set[y = 1 \vee y = 0]{(x, y) \in \realn^2} \\ + \closure{A} &= \set[0 \le y \le 1]{(x, y) \in \realn^2} + \end{align*} +\end{eg} + +\begin{rem} + \begin{enumerate}[(i)] + \item $\interior{A} \subset A$ + \item Boundary points of $A$ can be elements of $A$ or not. + \item $A \subset \interior{A} \cup \boundary{A}$, $~~\interior{A} \cap \boundary{A} = \varnothing$ + \item $\boundary{A} = \boundary{X \setminus A}$ + \end{enumerate} +\end{rem} + +\begin{thm} + Let $\metric$ be a metric space, $A \subset X$ and $x$ an interior point or boundary point of $A$. Then + \[ + \exists \anyseqdef{A}: ~~x_n \conv{} x + \] +\end{thm} +\begin{proof} + If $x \in A$ then this is trivial, so let $x \notin A$. Then + \begin{equation} + \forall n \in \natn ~\exists x_n \in \left(\oball[\frac{1}{n}](x) \cap A \ne \varnothing\right) + \end{equation} + We need to show that $\seq{x}$ converges to $x$. + \begin{equation} + \forall \epsilon > 0 ~\epsilon N \in \natn: ~~\frac{1}{N} < \epsilon + \end{equation} + For $n \ge N$ we have + \begin{equation} + \frac{1}{n} \le \frac{1}{N} < \epsilon + \end{equation} + and thus + \begin{equation} + d(x_n, x) < \frac{1}{n} < \epsilon + \end{equation} +\end{proof} + +\begin{defi}[Open and Closed sets] + Let $\metric$ be a metric space. $A \subset X$ is said to be + \begin{enumerate}[(i)] + \item open, if every point in $A$ is an interior point + \item closed, if $A$ contains all its boundary point + \item neighbourhood of $x \in A$, if $x$ is an interiot point of $A$ + \end{enumerate} +\end{defi} + +\begin{thm} + Let $\metric$ be a metric space and $A \subset X$. + \[ + A \text{ open} \iff X \setminus A \text{ closed} + \] +\end{thm} +\begin{proof} + \begin{subequations} + \begin{align} + A \text{ open} &\iff \forall x \in A: ~~x \in \interior{A} \\ + &\iff \forall x \in A: ~~x \in \boundary{A} \\ + &\iff X \setminus A \text{ contains all boundary point of } A \\ + &\iff X \setminus A \text{ contains all boundary points of } X \setminus A \\ + &\iff X \setminus A \text{ closed} + \end{align} + \end{subequations} +\end{proof} + +\begin{rem} + That doesn't mean $A$ has to be either open and closed. +\end{rem} + +\begin{eg} + Let $\metric$ be a metric space, $x \in X$ and $r > 0$. Then + \begin{align*} + \Oball(x) &= \set[d(x, y) < r]{y \in X} \text{ is open} \\ + \Cball(x) &= \set[d(x, y) < r]{y \in X} \text{ is closed} + \end{align*} +\end{eg} + +\begin{rem} + Consider the special case $a, b \in \realn$ with $a < b$ + \begin{align*} + (a, b) &= \oball[\frac{b - a}{2}]\left(\frac{a + b}{2}\right) \text{ open} \\ + [a, b] &= \cball[\frac{b - a}{2}]\left(\frac{a + b}{2}\right) \text{ closed} + \end{align*} +\end{rem} + +\begin{thm} + Let $\metric$ be a metric space and $A \subset X$. + \[ + A \text{ closed} \iff \forall \anyseqdef{A} \text{ convergent}: ~~\limn{x_n} \in A + \] +\end{thm} +\begin{proof} + Assume $A$ is closed. Let $\anyseqdef{A}$ be convergent to $x$. then + \begin{equation} + \forall \epsilon > 0 ~\exists N \in \natn: ~~x_n \in \oball(x) ~~\forall n \ge N + \end{equation} + This means that every $\epsilon$-ball around $x$ contains at least one point from $A$. + I.e. $x$ is always a point (or a boundary point) of $A$. From $A$ closed follows $x \in A$. + + Now assume $x \in \boundary{A}$. Then + \begin{equation} + \exists \anyseqdef{A}: ~~\seq{x} \conv{} x + \end{equation} + According to the prerequisites, $x \in A$. +\end{proof} + +\begin{thm} + Let $\metric$ be a metric space, and $\tau$ the set of all open subsets. Then + \begin{enumerate}[(i)] + \item $\varnothing \in \tau$, $~~X \in \tau$ + \item The union of any number of sets from $\tau$ is an open set + \[ + \bigcup_{t \in \tau} t \in \tau + \] + \item The intersection of finitely many sets from $\tau$ is an open set + \[ + \bigcap_{t \in \tau} t \in \tau + \] + \end{enumerate} +\end{thm} +\begin{proof} + \reader +\end{proof} + +\begin{rem} + \begin{enumerate}[(i)] + \item $\tau$ is said to be the topology induced by $d$ + \item \begin{itemize} + \item $\varnothing$, $X$ are also closed + \item The intersection of any number of closed sets is closed + \item The union of finitely many closed sets is closed + \end{itemize} + \item Infinitely many intersections of open sets are not open in general. + \end{enumerate} +\end{rem} + +\begin{thm} + Let $\metric$ be a metric space and $A \subset X$. Then + \[ + \interior{A} \text{ open} \implies \boundary{A}, \closure{A} \text{ closed} + \] +\end{thm} +\begin{proof} + Let $\interior{A}$ be open and $x \in \interior{A} \subset A$. This means + \begin{equation} + \exists \epsilon > 0: ~~\oball(x) \subset A + \end{equation} + We have to show that $\oball(x) \subset \interior{A}$. Let $y \in \oball(x)$. + Since $\oball(x)$ is open + \begin{equation} + \exists \delta > 0: ~~\oball[\delta](y) \subset \oball(x) \subset A + \end{equation} + This means that $y \in \oball(x)$ is interior point $A$. I.e. $\subset(x) \subset \interior{A}$, and thus $x$ is interior point of $\interior{A}$. + + Let $B = X \setminus A$. Then $\boundary{A} = \boundary{B}$ + \begin{equation} + X = A \cup B = \interior{A} \cup \boundary{A} \cup \interior{B} \cup \boundary{B} = \interior{A} \cup \boundary{A} \cup \interior{B} + \end{equation} + Then + \begin{subequations} + \begin{align} + A \text{ and } B \text{ are disjoint } &\implies \interior{A}, \interior{B} \text{ disjoint} \\ + &\implies \boundary{A} \text{ disjoint to } \interior{A}, \interior{B} + \end{align} + \end{subequations} + This results in + \begin{equation} + \boundary{A} = X \setminus (\underbrace{\interior{A} \cup \interior{B}}_{\text{open}}) \implies \boundary{A} \text{ closed} + \end{equation} + and + \begin{equation} + \closure{A} = A \cup \boundary{A} = \interior{A} \cup \boundary{A} = X \setminus \interior{B} \text{ closed} + \end{equation} +\end{proof} + +\begin{thm} + Let $\metric$ be a metric space and $A \subset X$ + \begin{align*} + \bigcup_{\substack{O \text{ open} \\ O \subset A}} O = \interior{A} && \text{and} && \bigcap_{\substack{C \text{ closed} \\ A \subset C}} C = \closure{A} + \end{align*} +\end{thm} +\begin{proof} + Let $\interior{A}$ is open and $\interior{A} \subset A$ + \begin{equation} + \implies \bigcup_{O \subset A \text{ open}} \supset \interior{A} + \end{equation} + Now let $O \subset A$ be open and $x \in O$, i.e. + \begin{equation} + \exists \epsilon > 0: ~~\oball(x) \subset O \subset A \implies x \in \interior{A} + \end{equation} + This implies that $O \subset \interior{A}$. Since this holds for all open $O \subset A$, this statement is proven. + The other statement follows from the complement. +\end{proof} + +\begin{thm} + Let $\metric$ be a complete space and $A \subset X$ be closed. Then $(A, d_A)$ is complete. +\end{thm} +\begin{proof} + \reader +\end{proof} + +\begin{rem} + Topological terms (open, closed, continuous, compact) don't just depend on $A$, but also on $X$. +\end{rem} + +\begin{defi} + Let $\metric$ be a metric space and $x \in X$. + \begin{enumerate}[(i)] + \item $x$ is said to be an isolated point if $\exists \epsilon > 0$ such that $\oball(x) = \set{x}$. + \item $x$ is said to be a limit point if it's not an isolated point. + \end{enumerate} +\end{defi} + +\begin{defi}[Punctured neighbourhood, Punctured ball] + $\dot{U} \subset X$ is said to be a punctured neighbourhood, if there is a neighbourhood $U$ of $x$ with $\dot{U} = U \setminus \set{x}$ + + A punctured ball is $\dot{\oball}(x) = \oball \setminus \set{x}$. +\end{defi} + +\begin{defi}[Limit of mappings] + Let $(X, d_X), (Y, d_Y)$ and $x$ a limit point of $X$. Let $\dot{U}$ be a punctured neighbourhood of $x$ and $f: \dot{U} \rightarrow Y$. + Then $f$ converges to $y \in Y$ in $x$ ($y$ is said to be the limit of $f$ in $x$), if + \[ + \forall \epsilon > 0 ~\exists \delta > 0: ~~f(\tilde{x}) \in \oball(y) ~~[d(f(\tilde{x}), y) < \epsilon] + \] + if $\tilde{x} \in \dot{\oball}(x)$ [$d(\tilde{x}, x) < \delta$] +\end{defi} + +\begin{eg} + Let $f, g: \realn^2 \setminus \set{0} \rightarrow \realn$. + \begin{align*} + f(x) := \norm{x}^2 && g(x) := \frac{1}{\norm{x}} + \end{align*} + Then $\limes{x}{0} f(x) = 0$, because for $\epsilon > 0$ and $\delta = \sqrt{\epsilon}$ we have + \[ + d(\tilde{x}, 0) = \norm{\tilde{x} - 0} = \tilde{x} < \delta \implies d(f(\tilde{x}), 0) = \abs{\norm{\tilde{x}}^2 - 0} = \norm{\tilde{x}}^2 < \epsilon = \delta^2 + \] +\end{eg} + +\begin{thm} + \[ + f \text{ converges to } y \in Y \text{ in } x \iff \forall \anyseqdef{X}: ~~f(x_n) \conv{x_n \rightarrow x} y + \] +\end{thm} +\begin{proof} + Let $\anyseqdef{X}$ with $x_n \conv{} x$. Let $\epsilon > 0$, then + \begin{equation} + \exists \delta > 0: ~~f(\tilde{x}) \in \oball(y) \text{ if } \tilde{x} \in \oball[\delta](x) + \end{equation} + Furthermore + \begin{equation} + \exists N \in \natn: ~~x_n \in \oball[\delta](x) ~~\forall n \ge N + \end{equation} + Then + \begin{equation} + f(x_n) \in \oball(y) ~~\forall n \ge N + \end{equation} + To prove the other direction, assume $f$ doesn't converge to $y$ in $y$. This means + \begin{equation} + \exists \epsilon > 0: ~~\exists \tilde{x} \in \oball[\delta](x) \text{ but } f(\tilde{x}) \notin \oball(y) ~~\forall \delta > 0 + \end{equation} + Therefore + \begin{equation} + \forall n \in \natn: ~~\exists x_n \in \oball[\frac{1}{n}](x) + \end{equation} + We know that $x_n \conv{} x$ since $d(x_n, x) < \frac{1}{n}$, but $f(x_n)$ doesn't converge to $y$ since $d(f(x_n), y) \ge \epsilon$. +\end{proof} + +\begin{cor} + Let $\metric$ be a metric space, $x \in X$ a limit point and $\dot{U}$ a punctured neighbourhood of $x$. Let $f, g: \dot{U} \rightarrow \field$ with + \begin{align*} + \limes{\tilde{x}}{x} f(\tilde{x}) = y_1 && \limes{\tilde{x}}{x} g(\tilde{x}) = y_2 + \end{align*} + Then + \begin{align*} + \limes{\tilde{x}}{x} (f + g)(\tilde{x}) = y_1 + y_2 && \limes{\tilde{x}}{x} (f \cdot g)(\tilde{x}) = y_1 \cdot y_2 \\ + \limes{\tilde{x}}{x} \left(\frac{f}{g}\right)(\tilde{x}) = \frac{y_1}{y_2} + \end{align*} +\end{cor} +\begin{hproof} + Draw parallels back to number sequences +\end{hproof} +\end{document} \ No newline at end of file diff --git a/chapters/topo_of_metr_spaces.tex b/chapters/topo_of_metr_spaces.tex index b6e1af2..d7ee84d 100644 --- a/chapters/topo_of_metr_spaces.tex +++ b/chapters/topo_of_metr_spaces.tex @@ -6,4 +6,5 @@ \subfile{sections/metr_and_normed.tex} \subfile{sections/seq_ser_limits.tex} + \subfile{sections/open_closed_sets.tex} \end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index 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\newcommand{\series}[2][\infty]{\sum_{#2 = 1}^{#1}} \newcommand{\finite}{\text{ finite}} \newcommand{\conj}[1]{\overline{#1}} -\newcommand{\closure}[1]{\overline{#1}} \newcommand{\must}[1][=]{\stackrel{!}{#1}} \newcommand{\metric}[1][X]{(#1, d)} @@ -72,6 +71,10 @@ \newcommand{\Oball}{\oball[r]} \newcommand{\Cball}{\cball[r]} +\newcommand{\interior}[1]{\mathring{#1}} +\newcommand{\boundary}[1]{\partial #1} +\newcommand{\closure}[1]{\bar{#1}} + \newcommand{\conv}[1]{\xrightarrow{\makebox[2em][c]{$\scriptstyle#1$}}} \newcommand{\convinf}{\conv{n \rightarrow \infty}}