4.2: Open Sets
- Page ID
- 22658
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We say a set \(U \subset \mathbb{R}\) is open if for every \(x \in U\) there exists \(\epsilon>0\) such that
\[(x-\epsilon, x+\epsilon) \subset U.\]
Every open interval \(I\) is an open set.
- Proof
-
Suppose \(I=(a, b),\) where \(a<b\) are extended real numbers. Given \(x \in I,\) let \(\epsilon\) be the smaller of \(x-a\) and \(b-x .\) Suppose \(y \in(x-\epsilon, x+\epsilon) .\) If \(b=+\infty,\) then \(b>y ;\) otherwise, we have
\[b-y>b-(x+\epsilon)=(b-x)-\epsilon \geq(b-x)-(b-x)=0,\]
so \(b>y .\) If \(a=-\infty,\) then \(a<y ;\) otherwise,
\[y-a>(x-\epsilon)-a=(x-a)-\epsilon \geq(x-a)-(x-a)=0,\]
so \(a<y .\) Thus \(y \in I\) and \(I\) is an open set. \(\quad\) Q.E.D.
Note that \(\mathbb{R} \text { is an open set (it is, in fact, the open interval }(-\infty,+\infty)),\) as is \(\emptyset\) (it satisfies the definition trivially).
Suppose \(A\) is a set and, for each \(\alpha \in A, U_{\alpha}\) is an open set. Then
\[\bigcup_{\alpha \in A} U_{\alpha}\]
is an open set.
- Proof
-
Let \(x \in \cup_{\alpha \in A} U_{\alpha} .\) Then \(x \in U_{\alpha}\) for some \(\alpha \in A .\) Since \(U_{\alpha}\) is open, there exists an \(\epsilon>0\) such that \((x-\epsilon, x+\epsilon) \subset U_{\alpha} .\) Thus
\[(x-\epsilon, x+\epsilon) \subset U_{\alpha} \subset \bigcup_{\alpha \in A} U_{\alpha}.\]
Hence \(\bigcup_{\alpha \in A} U_{\alpha}\) is open. \(\quad\) Q.E.D.
Suppose \(U_{1}, U_{2}, \ldots, U_{n}\) is a finite collection of open sets. Then
\[\bigcap_{i=1}^{n} U_{i}\]
is open.
- Proof
-
Let \(x \in \bigcap_{i=1}^{n} U_{i} .\) Then \(x \in U_{i}\) for every \(i=1,2, \ldots, n .\) For each \(i\), choose \(\epsilon_{i}>0\) such that \(\left(x-\epsilon_{i}, x+\epsilon_{i}\right) \subset U_{i} .\) Let \(\epsilon\) be the smallest of \(\epsilon_{1}, \epsilon_{2}, \ldots, \epsilon_{n} .\) Then \(\epsilon>0\) and
\[(x-\epsilon, x+\epsilon) \subset\left(x-\epsilon_{i}, x+\epsilon_{i}\right) \subset U_{i}\]
for every \(i=1,2, \ldots, n .\) Thus
\[(x-\epsilon, x+\epsilon) \subset \bigcap_{i=1}^{n} U_{i}.\]
Hence \(\bigcap_{i=1}^{n} U_{i}\) is an open set. \(\quad\) Q.E.D.
Let \(A \subset \mathbb{R} .\) We say \(x \in A\) is an interior point of \(A\) if there exists an \(\epsilon>0\) such that \((x-\epsilon, x+\epsilon) \subset A .\) We call the set of all interior points of \(A\) the interior of \(A,\) denoted \(A^{\circ} .\)
Show that if \(A \subset \mathbb{R},\) then \(A^{\circ}\) is open.
Show that \(A\) is open if and only if \(A=A^{\circ}\).
Let \(U \subset \mathbb{R}\) be a nonempty open set. Show that sup \(U \notin U\) and \(\inf U \notin U\).