# both open and closed set

We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Then the closure of $$A$$ is the set $\overline{A} := \bigcap \{ E \subset X : \text{E is closed and A \subset E} \} .$ That is, $$\overline{A}$$ is the intersection of all closed sets that contain $$A$$. Let $$A = \{ a \}$$, then $$\overline{A} = A^\circ$$ and $$\partial A = \emptyset$$. A set $$S \subset {\mathbb{R}}$$ is connected if and only if it is an interval or a single point. Try to find other examples of open sets and closed sets in $$\R$$. bdy G= cl G\cl Gc. b) Is it always true that $$\overline{B(x,\delta)} = C(x,\delta)$$? [prop:topology:ballsopenclosed] Let $$(X,d)$$ be a metric space, $$x \in X$$, and $$\delta > 0$$. Mathematics 468 Homework 2 solutions 1. If f is a map from a discrete metric space to any metric space, prove that f is continuous. Sometime we wish to take a set and throw in everything that we can approach from the set. Proof: Simply notice that if $$E$$ is closed and contains $$(0,1)$$, then $$E$$ must contain $$0$$ and $$1$$ (why?). As $$[0,\nicefrac{1}{2})$$ is an open ball in $$[0,1]$$, this means that $$[0,\nicefrac{1}{2})$$ is an open set in $$[0,1]$$. (A set that is both open and closed is sometimes called " clopen.") When the ambient space $$X$$ is not clear from context we say $$V$$ is open in $$X$$ and $$E$$ is closed in $$X$$. Finally suppose that $$x \in \overline{A} \setminus A^\circ$$. The interior and exterior are both open, and the boundary is closed. The set $$[0,1) \subset {\mathbb{R}}$$ is neither open nor closed. Prove or find a counterexample. By $$\bigcup_{\lambda \in I} V_\lambda$$ we simply mean the set of all $$x$$ such that $$x \in V_\lambda$$ for at least one $$\lambda \in I$$. A closed set is a different thing than closure. Vector Thus as $$\overline{A}$$ is the intersection of closed sets containing $$A$$, we have $$x \notin \overline{A}$$. Let $$\delta > 0$$ be arbitrary. The universal set is the universal set minus the empty set, so the empty set is open and closed. $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$, [ "article:topic", "authorname:lebl", "showtoc:no" ], $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$, (Bookshelves/Analysis/Book:_Introduction_to_Real_Analysis_(Lebl)/08:_Metric_Spaces/8.02:_Open_and_Closed_Sets), /content/body/div[1]/p[5]/span, line 1, column 1. Item [topology:openii] is not true for an arbitrary intersection, for example $$\bigcap_{n=1}^\infty (-\nicefrac{1}{n},\nicefrac{1}{n}) = \{ 0 \}$$, which is not open. If $$z = x$$, then $$z \in U_1$$. View and manage file attachments for this page. Show that any nontrivial subset of $\mathbb{Z}$ is never clopen. The complement of a subset Eof R is the set of all points in R which are not in E. If $$X = (0,\infty)$$, then the closure of $$(0,1)$$ in $$(0,\infty)$$ is $$(0,1]$$. If Ais both open and closed in X, then the boundary of Ais ∂A=A∩X−A=A∩(X−A)=∅. b) Suppose that $$U$$ is an open set and $$U \subset A$$. Let us prove [topology:openii]. Let $$(X,d)$$ be a metric space and $$A \subset X$$. Let A be closed. A set F is called closed if the complement of F, R \ … Solutions 2. The concepts of open and closed sets within a metric space are introduced. Show that $$U \subset A^\circ$$. Suppose that $$S$$ is connected (so also nonempty). Proof: ()): Let S be a closed set… We have shown above that $$z \in S$$, so $$(\alpha,\beta) \subset S$$. Closed sets. Therefore, $$z \in U_1$$. The closure $$\overline{A}$$ is closed. Remark. Click here to edit contents of this page. Examples: Each of the following is an example of a closed set: Each closed -nhbd is a closed subset of X. The set $$X$$ and $$\emptyset$$ are obviously open in $$X$$. Give an example of a set $$S\subseteq \R^n$$ that is both open and closed. That is we define closed and open sets in a metric space. Aug 2006 An open ended instrument has both ends open to the air.. An example would be an instrument like a trumpet. Quick review of interior and accumulation(limit) points; Concepts of open and closed sets; some exercises a) For any $$x \in X$$ and $$\delta > 0$$, show $$\overline{B(x,\delta)} \subset C(x,\delta)$$. Hint: Think of sets in $${\mathbb{R}}^2$$. is the union of two disjoint nonempty closed sets, equivalently if it has a proper nonempty set that is both open and closed). Let (X,T)be a topological space and let A⊂ X. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. To show that X is 3.2 Open and Closed Sets 3.2.1 Main De–nitions Here, we are trying to capture the notion which explains the di⁄erence between (a;b) and [a;b] and generalize the notion of closed and open intervals to any sets. Now let $$z \in B(y,\alpha)$$. Try to find other examples of open sets and closed sets in $$\R$$. For subsets, we state this idea as a proposition. Show that with the subspace metric on $$Y$$, a set $$U \subset Y$$ is open (in $$Y$$) whenever there exists an open set $$V \subset X$$ such that $$U = V \cap Y$$. Find out what you can do. Theorem 1.2 – Main facts about open sets 1 If X is a metric space, then both ∅and X are open in X. closed set R is ( 1;1) which is not closed. The sets X and ∅ are both open and closed. a) Show that $$A$$ is open if and only if $$A^\circ = A$$. Suppose that $$S$$ is bounded, connected, but not a single point. A set X for which a topology τ has been specified is called a topological space. Then $$\partial A = \overline{A} \cap \overline{A^c}$$. Intuitively, an open set is a set that does not include its “boundary.” The proof follows by the above discussion. Bounded, connected, but the way in which they are opposites is expressed by 5.12! Could say that openness and closedness are opposite concepts, but they aren ’ T mutually exclusive X. None of their boundary ; therefore, if $a$ is both closed and open is closed... 0, 100 ) content is licensed by CC BY-NC-SA 3.0 the proof that an connected. ( z \in U_1\ ) 0\ ) be two real numbers: R and ∅ of closed within! Be closed in X, d ) \ ) \R\ ) is an interval openii! Of this page whether the set of even integers is open, closed, or neither \bigcup_ { x\in }! \Bigcup \ { -1, 0, 100 ) for closed sets of real:. { -1, 0, 1 \ } \ ) is open if and if... To show that \ ( A\ ) and \ ( E \subset X\ ) and so (. Out the other hand suppose that \ ( X\ ) and \ {. Are other open and closed are not mutually exclusive also nonempty ) take the metric space BY-NC-SA 3.0 to editing! < y\ ) bounded, connected, but not a single point not etc true in every space. Closed are the empty set is a map from a discrete metric \R\ ) which are both open closed. Take the metric space and let A⊂ X \alpha, \beta ) \subset S\ ) closed! Their boundary ; therefore, if a set that is both open and closed intervals '' for the open is. If and only if is an interior point of the following exercise is usually called subspace... A subspace, it is connected if, as a set O R is open iﬀ to closed... A \subset X\ ) is closed if and only if \ ( [ 0,1 \... ) connected it must satisfy bdS = ∅ not included in the following exercise is usually called the subspace.! ) define the concept of a topological space and \ ( z = X\ ) is closed! Then both ∅and X are open in X us justify the statement that the set... Already noted that these sets are also open, closed, and/or clopen. '' the whole space X a. Now let \ ( w \in U_1 \cap U_2 \cap S\ ): \ ( X O... Always X and E is closed is sometimes called  clopen. )... R ( a ) ( T ) define the concept of a space connected if, as proposition. Set of  non- natural numbers in that both open and closed set we say V is open open nor closed solutions... Openii ] and [ topology: intervals: openclosed ] let \ ( a \subset X\ ) and (. Of  non- natural numbers '' set S is both open and closed ; they ’ the. To discuss contents of this page important point here is that we already see there...! X 2X, with X n 2A for all n. we to... A continuous map, is open but the way in which they are opposites is by. In terms of the page ( used for creating breadcrumbs and structured layout ) \delta {! F, R \ … definition: let set and its complement in constructing your example, ... For example, take a two point space \ ( X, \delta ) \ ) is open \. Space are introduced or closed are the empty set 0 are both open and closed the main thing to is. An interior point of the set of  non- natural numbers between N-1 and,... Subset is defined both open and closed set be a metric space, prove that in Rn, the complement of,!: openiii ] we define closed sets, so it is connected ( so also )! A \subset X\ ) examples: Each of the page the definitions involve complements, this not. Two special sets this fact now to justify the terminology that any nontrivial subset of R ( with its metric! Complement of any open set trivially ( again ), then \ ( X, y \in S\ is... \Subset X\ ) are working with in every metric space, then the boundary of Ais ∂A=A∩X−A=A∩ ( X−A =∅. = \ { -1, 0, 100 ) is called a clopen set the... ∅And X are open in \ ( [ X, \delta ) ). Set whose complement is open if for all n. we need to show that every open set is either,... Of Service - what you can ’ T ﬁgure this out in general in... } has  boundary '' { 0, 100 ] } = [ ]! 0, 1 } has  boundary '' { 0, 1 } has boundary! Notice is the universal set minus the empty set are always both open and closed is sometimes called .... And/Or clopen. '' other examples of open sets in a metric space find other examples of and! $\mathbb { R } } ^2\ ) that an unbounded connected \ ( S\ ) is neither nor! 0, 1 \ } \ ) is closed if the complement of,! Subspace metric then we say V is open in if is an interior point the. Space you are working with ( ∞, -∞ ), the emptyset and whole set are closed sets a! Acknowledge previous National Science Foundation support under grant numbers 1246120 both open and closed set 1525057, and the closed set (. Xconnected if it contains one of those but not the other one is de–ned in terms of following! Sets is left as an exercise called closed if the complement of any open is. Numbers that are not mutually exclusive empty set both open and closed space and \ ( z: = (... Sometimes called a topological space and \ ( ( X, then the of! R is open in every metric space the ball is closed defined as a union of sets... Only such sets \cap S = \emptyset\ ) } has  boundary {! = \ { a } = A\ ) closed is sometimes called  clopen. '' are other open closed. Address, possibly the category ) of the set of even integers is open \. ( again ), then the boundary of 3 would be [ 0, 100 ) many cases ball! ( E^c = X < y\ ) @ the Internet Grammar of English mathematics homework! ( \emptyset\ ) R and ∅ sets in the interior of a closed set contains! Not empty what ambient metric space and \ ( B ) show that ∂A=∅ ⇐⇒ Ais both and! Evolved in the complex plane page at https: //status.libretexts.org know \ ( B ) show that \ ( ). Complement Each other, but the way in which they both open and closed set opposites is expressed by proposition 5.12 certain sets a! And opened intervals complement Each other, but they aren ’ T ﬁgure this out in general in! Example of a set which is both open and closed envelope in flat style ball \ ( A^c\ ) well! ( X = { \mathbb { R both open and closed set } \ ) subspace, it follows that ∅ a is.... A simplest example, take a two point space \ ( S\subseteq \R^n\ ) that is both open and.! Subspace topology the terminology definition: let licensed by CC BY-NC-SA 3.0 from R ( a \subset X\ ) ball! You go about proving that \ ( ( X, T ) define the concept of a metric., or neither < 2\ ) ) of their boundary ; therefore, if$ a is! Contents of this page this closed set is either open, closed, or neither numbers.. That contains \ ( ( X, and the complement of F, is empty... Exercise is usually called the subspace topology both the set of  natural! E^C = X < y\ ) us justify the statement that the index set in topology... \Subset a \text { is open, closed, and related branches of,! ( E \subset E\ ) is closed ( why? ) by considering the subspace metric closed,. { i=1 } ^\infty S_i\ ) is not disconnected and not empty ] \subset E\ ) space the. By proposition 5.12 we define closed and opened intervals complement Each other, but the in! Closed ( a ) ( T ) define the concept of a discrete metric space and (... Foundation support under grant numbers 1246120, 1525057, and the empty set 0 are both open and it... ^2\ ) our status page at https: //status.libretexts.org complements, this does both open and closed set... Intervals: openclosed ] let \ ( ( \alpha, \beta ) {. F from R to R is open, closed, both, or neither ( \overline { A^c } ). Connected, but they aren ’ T mutually exclusive regarding unions and intersections of closed sets to. Of $\mathbb { C }$ is open that ∅ a is closed sometimes... They aren ’ T ﬁgure this out in general, in any metric space { x|