However, if you are considering singletons as subsets of a larger topological space, this will depend on the properties of that space. ), Are singleton set both open or closed | topology induced by metric, Lecture 3 | Collection of singletons generate discrete topology | Topology by James R Munkres. { Sign In, Create Your Free Account to Continue Reading, Copyright 2014-2021 Testbook Edu Solutions Pvt. {\displaystyle 0} Now lets say we have a topological space X in which {x} is closed for every xX. By accepting all cookies, you agree to our use of cookies to deliver and maintain our services and site, improve the quality of Reddit, personalize Reddit content and advertising, and measure the effectiveness of advertising. , Prove Theorem 4.2. If you preorder a special airline meal (e.g. The best answers are voted up and rise to the top, Not the answer you're looking for? {\displaystyle \iota } It only takes a minute to sign up. In the given format R = {r}; R is the set and r denotes the element of the set. It only takes a minute to sign up. The given set has 5 elements and it has 5 subsets which can have only one element and are singleton sets. . } We walk through the proof that shows any one-point set in Hausdorff space is closed. {\displaystyle \{A\}} and Tis called a topology Exercise. (Calculus required) Show that the set of continuous functions on [a, b] such that. S Locally compact hausdorff subspace is open in compact Hausdorff space?? If a law is new but its interpretation is vague, can the courts directly ask the drafters the intent and official interpretation of their law? The reason you give for $\{x\}$ to be open does not really make sense. In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed.That this is possible may seem counter-intuitive, as the common meanings of open and closed are antonyms, but their mathematical definitions are not mutually exclusive.A set is closed if its complement is open, which leaves the possibility of an open set whose complement . Has 90% of ice around Antarctica disappeared in less than a decade? ( The main stepping stone: show that for every point of the space that doesn't belong to the said compact subspace, there exists an open subset of the space which includes the given point, and which is disjoint with the subspace. But if this is so difficult, I wonder what makes mathematicians so interested in this subject. x If these sets form a base for the topology $\mathcal{T}$ then $\mathcal{T}$ must be the cofinite topology with $U \in \mathcal{T}$ if and only if $|X/U|$ is finite. in a metric space is an open set. Let X be a space satisfying the "T1 Axiom" (namely . The idea is to show that complement of a singleton is open, which is nea. In a discrete metric space (where d ( x, y) = 1 if x y) a 1 / 2 -neighbourhood of a point p is the singleton set { p }. Also, reach out to the test series available to examine your knowledge regarding several exams. The best answers are voted up and rise to the top, Not the answer you're looking for? For every point $a$ distinct from $x$, there is an open set containing $a$ that does not contain $x$. Assume for a Topological space $(X,\mathcal{T})$ that the singleton sets $\{x\} \subset X$ are closed. Equivalently, finite unions of the closed sets will generate every finite set. To show $X-\{x\}$ is open, let $y \in X -\{x\}$ be some arbitrary element. The Cantor set is a closed subset of R. To construct this set, start with the closed interval [0,1] and recursively remove the open middle-third of each of the remaining closed intervals . Why do small African island nations perform better than African continental nations, considering democracy and human development? Then every punctured set $X/\{x\}$ is open in this topology. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. {\displaystyle \{A,A\},} Anonymous sites used to attack researchers. Assume for a Topological space $(X,\mathcal{T})$ that the singleton sets $\{x\} \subset X$ are closed. Prove that in the metric space $(\Bbb N ,d)$, where we define the metric as follows: let $m,n \in \Bbb N$ then, $$d(m,n) = \left|\frac{1}{m} - \frac{1}{n}\right|.$$ Then show that each singleton set is open. The singleton set has only one element in it. The complement of is which we want to prove is an open set. Demi Singleton is the latest addition to the cast of the "Bass Reeves" series at Paramount+, Variety has learned exclusively. 3 . A subset C of a metric space X is called closed This topology is what is called the "usual" (or "metric") topology on $\mathbb{R}$. of is an ultranet in Consider $\ {x\}$ in $\mathbb {R}$. X {\displaystyle X} The cardinal number of a singleton set is one. I downoaded articles from libgen (didn't know was illegal) and it seems that advisor used them to publish his work, Brackets inside brackets with newline inside, Brackets not tall enough with smallmatrix from amsmath. Does Counterspell prevent from any further spells being cast on a given turn? Then $X\setminus \{x\} = (-\infty, x)\cup(x,\infty)$ which is the union of two open sets, hence open. { The Bell number integer sequence counts the number of partitions of a set (OEIS:A000110), if singletons are excluded then the numbers are smaller (OEIS:A000296). In a usual metric space, every singleton set {x} is closed #Shorts - YouTube 0:00 / 0:33 Real Analysis In a usual metric space, every singleton set {x} is closed #Shorts Higher. We will learn the definition of a singleton type of set, its symbol or notation followed by solved examples and FAQs. is a singleton whose single element is the closure of the set of even integers. Arbitrary intersectons of open sets need not be open: Defn so, set {p} has no limit points The power set can be formed by taking these subsets as it elements. It is enough to prove that the complement is open. Summing up the article; a singleton set includes only one element with two subsets. Let E be a subset of metric space (x,d). 968 06 : 46. Thus since every singleton is open and any subset A is the union of all the singleton sets of points in A we get the result that every subset is open. What are subsets of $\mathbb{R}$ with standard topology such that they are both open and closed? There is only one possible topology on a one-point set, and it is discrete (and indiscrete). If so, then congratulations, you have shown the set is open. Having learned about the meaning and notation, let us foot towards some solved examples for the same, to use the above concepts mathematically. $\emptyset$ and $X$ are both elements of $\tau$; If $A$ and $B$ are elements of $\tau$, then $A\cap B$ is an element of $\tau$; If $\{A_i\}_{i\in I}$ is an arbitrary family of elements of $\tau$, then $\bigcup_{i\in I}A_i$ is an element of $\tau$. This states that there are two subsets for the set R and they are empty set + set itself. Every set is an open set in . How can I find out which sectors are used by files on NTFS? Set Q = {y : y signifies a whole number that is less than 2}, Set Y = {r : r is a even prime number less than 2}. Singleton Set has only one element in them. In axiomatic set theory, the existence of singletons is a consequence of the axiom of pairing: for any set A, the axiom applied to A and A asserts the existence of is a principal ultrafilter on In the space $\mathbb R$,each one-point {$x_0$} set is closed,because every one-point set different from $x_0$ has a neighbourhood not intersecting {$x_0$},so that {$x_0$} is its own closure. ), von Neumann's set-theoretic construction of the natural numbers, https://en.wikipedia.org/w/index.php?title=Singleton_(mathematics)&oldid=1125917351, The statement above shows that the singleton sets are precisely the terminal objects in the category, This page was last edited on 6 December 2022, at 15:32. { All sets are subsets of themselves. y This topology is what is called the "usual" (or "metric") topology on $\mathbb{R}$. 0 The cardinality (i.e. In summary, if you are talking about the usual topology on the real line, then singleton sets are closed but not open. In $\mathbb{R}$, we can let $\tau$ be the collection of all subsets that are unions of open intervals; equivalently, a set $\mathcal{O}\subseteq\mathbb{R}$ is open if and only if for every $x\in\mathcal{O}$ there exists $\epsilon\gt 0$ such that $(x-\epsilon,x+\epsilon)\subseteq\mathcal{O}$. X {\displaystyle X} What age is too old for research advisor/professor? one. Some important properties of Singleton Set are as follows: Types of sets in maths are important to understand the theories in maths topics such as relations and functions, various operations on sets and are also applied in day-to-day life as arranging objects that belong to the alike category and keeping them in one group that would help find things easily. : The following result introduces a new separation axiom. Consider $\{x\}$ in $\mathbb{R}$. Each of the following is an example of a closed set. Then, $\displaystyle \bigcup_{a \in X \setminus \{x\}} U_a = X \setminus \{x\}$, making $X \setminus \{x\}$ open. Show that every singleton in is a closed set in and show that every closed ball of is a closed set in . The cardinal number of a singleton set is 1. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. in Proof: Let and consider the singleton set . I . Since all the complements are open too, every set is also closed. When $\{x\}$ is open in a space $X$, then $x$ is called an isolated point of $X$. Why are physically impossible and logically impossible concepts considered separate in terms of probability? X x. However, if you are considering singletons as subsets of a larger topological space, this will depend on the properties of that space. PS. x Why do universities check for plagiarism in student assignments with online content? 690 14 : 18. The following holds true for the open subsets of a metric space (X,d): Proposition X x This is definition 52.01 (p.363 ibid. Within the framework of ZermeloFraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. A set with only one element is recognized as a singleton set and it is also known as a unit set and is of the form Q = {q}. Different proof, not requiring a complement of the singleton. So $r(x) > 0$. Every singleton is compact. Can I tell police to wait and call a lawyer when served with a search warrant? Then every punctured set $X/\{x\}$ is open in this topology. With the standard topology on R, {x} is a closed set because it is the complement of the open set (-,x) (x,). Example 1: Find the subsets of the set A = {1, 3, 5, 7, 11} which are singleton sets. y For $T_1$ spaces, singleton sets are always closed. Note. @NoahSchweber:What's wrong with chitra's answer?I think her response completely satisfied the Original post. $y \in X, \ x \in cl_\underline{X}(\{y\}) \Rightarrow \forall U \in U(x): y \in U$, Singleton sets are closed in Hausdorff space, We've added a "Necessary cookies only" option to the cookie consent popup. You may want to convince yourself that the collection of all such sets satisfies the three conditions above, and hence makes $\mathbb{R}$ a topological space. rev2023.3.3.43278. If these sets form a base for the topology $\mathcal{T}$ then $\mathcal{T}$ must be the cofinite topology with $U \in \mathcal{T}$ if and only if $|X/U|$ is finite. How much solvent do you add for a 1:20 dilution, and why is it called 1 to 20? Every singleton set is closed. rev2023.3.3.43278. Ranjan Khatu. I am afraid I am not smart enough to have chosen this major. The singleton set has only one element in it. The rational numbers are a countable union of singleton sets. Get Daily GK & Current Affairs Capsule & PDFs, Sign Up for Free x Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. If you are working inside of $\mathbb{R}$ with this topology, then singletons $\{x\}$ are certainly closed, because their complements are open: given any $a\in \mathbb{R}-\{x\}$, let $\epsilon=|a-x|$. That is, why is $X\setminus \{x\}$ open? Learn more about Stack Overflow the company, and our products. for X. Then $(K,d_K)$ is isometric to your space $(\mathbb N, d)$ via $\mathbb N\to K, n\mapsto \frac 1 n$. What does that have to do with being open? Defn called open if, Already have an account? denotes the singleton Anonymous sites used to attack researchers. Since X\ {$b$}={a,c}$\notin \mathfrak F$ $\implies $ In the topological space (X,$\mathfrak F$),the one-point set {$b$} is not closed,for its complement is not open. Follow Up: struct sockaddr storage initialization by network format-string, Acidity of alcohols and basicity of amines. So in order to answer your question one must first ask what topology you are considering. Sets in mathematics and set theory are a well-described grouping of objects/letters/numbers/ elements/shapes, etc. x Why does [Ni(gly)2] show optical isomerism despite having no chiral carbon? A set containing only one element is called a singleton set. Answer (1 of 5): You don't. Instead you construct a counter example. The singleton set has only one element, and hence a singleton set is also called a unit set. This implies that a singleton is necessarily distinct from the element it contains,[1] thus 1 and {1} are not the same thing, and the empty set is distinct from the set containing only the empty set. So in order to answer your question one must first ask what topology you are considering. of X with the properties. Compact subset of a Hausdorff space is closed. If there is no such $\epsilon$, and you prove that, then congratulations, you have shown that $\{x\}$ is not open. So: is $\{x\}$ open in $\mathbb{R}$ in the usual topology? Metric Spaces | Lecture 47 | Every Singleton Set is a Closed Set, Singleton sets are not Open sets in ( R, d ), Every set is an open set in discrete Metric Space, Open Set||Theorem of open set||Every set of topological space is open IFF each singleton set open, The complement of singleton set is open / open set / metric space. This parameter defaults to 'auto', which tells DuckDB to infer what kind of JSON we are dealing with.The first json_format is 'array_of_records', while the second is . : } Why does [Ni(gly)2] show optical isomerism despite having no chiral carbon? Proposition They are also never open in the standard topology. Thus, a more interesting challenge is: Theorem Every compact subspace of an arbitrary Hausdorff space is closed in that space. , i.e. Let $F$ be the family of all open sets that do not contain $x.$ Every $y\in X \setminus \{x\}$ belongs to at least one member of $F$ while $x$ belongs to no member of $F.$ So the $open$ set $\cup F$ is equal to $X\setminus \{x\}.$. Are Singleton sets in $\mathbb{R}$ both closed and open? Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Share Cite Follow edited Mar 25, 2015 at 5:20 user147263 The following are some of the important properties of a singleton set. Singleton sets are not Open sets in ( R, d ) Real Analysis. In this situation there is only one whole number zero which is not a natural number, hence set A is an example of a singleton set. Let $(X,d)$ be a metric space such that $X$ has finitely many points. Contradiction. Theorem 17.8. Each open -neighborhood NOTE:This fact is not true for arbitrary topological spaces. ncdu: What's going on with this second size column? {\displaystyle \{x\}} But I don't know how to show this using the definition of open set(A set $A$ is open if for every $a\in A$ there is an open ball $B$ such that $x\in B\subset A$). So: is $\{x\}$ open in $\mathbb{R}$ in the usual topology? Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Title. Define $r(x) = \min \{d(x,y): y \in X, y \neq x\}$. in X | d(x,y) < }. For a set A = {a}, the two subsets are { }, and {a}. The singleton set has two sets, which is the null set and the set itself. $y \in X, \ x \in cl_\underline{X}(\{y\}) \Rightarrow \forall U \in U(x): y \in U$. By the Hausdorff property, there are open, disjoint $U,V$ so that $x \in U$ and $y\in V$. Find the closure of the singleton set A = {100}. You may just try definition to confirm. Why do many companies reject expired SSL certificates as bugs in bug bounties? The subsets are the null set and the set itself. You can also set lines='auto' to auto-detect whether the JSON file is newline-delimited.. Other JSON Formats. Reddit and its partners use cookies and similar technologies to provide you with a better experience. Solution:Let us start checking with each of the following sets one by one: Set Q = {y: y signifies a whole number that is less than 2}. The singleton set is of the form A = {a}. set of limit points of {p}= phi Then $X\setminus \ {x\} = (-\infty, x)\cup (x,\infty)$ which is the union of two open sets, hence open. Are there tables of wastage rates for different fruit and veg? 690 07 : 41. } Connect and share knowledge within a single location that is structured and easy to search. In summary, if you are talking about the usual topology on the real line, then singleton sets are closed but not open. Then the set a-d<x<a+d is also in the complement of S. . The set {y If using the read_json function directly, the format of the JSON can be specified using the json_format parameter. The two possible subsets of this singleton set are { }, {5}. They are also never open in the standard topology. Ummevery set is a subset of itself, isn't it? A singleton set is a set containing only one element. empty set, finite set, singleton set, equal set, disjoint set, equivalent set, subsets, power set, universal set, superset, and infinite set. {\displaystyle \{0\}} {\displaystyle X.}. Every singleton set is an ultra prefilter. Suppose Y is a Since a singleton set has only one element in it, it is also called a unit set. Each closed -nhbd is a closed subset of X. x Generated on Sat Feb 10 11:21:15 2018 by, space is T1 if and only if every singleton is closed, ASpaceIsT1IfAndOnlyIfEverySingletonIsClosed, ASpaceIsT1IfAndOnlyIfEverySubsetAIsTheIntersectionOfAllOpenSetsContainingA. There are various types of sets i.e. Singleton sets are open because $\{x\}$ is a subset of itself. The null set is a subset of any type of singleton set. Well, $x\in\{x\}$. there is an -neighborhood of x 968 06 : 46. The powerset of a singleton set has a cardinal number of 2. denotes the class of objects identical with of d to Y, then. Since the complement of $\{x\}$ is open, $\{x\}$ is closed. {\displaystyle X.} } In mathematics, a singleton, also known as a unit set[1] or one-point set, is a set with exactly one element. um so? Here's one. Definition of closed set : What is the point of Thrower's Bandolier? Open Set||Theorem of open set||Every set of topological space is open IFF each singleton set open . My question was with the usual metric.Sorry for not mentioning that. number of elements)in such a set is one. Example: Consider a set A that holds whole numbers that are not natural numbers. How many weeks of holidays does a Ph.D. student in Germany have the right to take? What to do about it? metric-spaces. How much solvent do you add for a 1:20 dilution, and why is it called 1 to 20? In particular, singletons form closed sets in a Hausdor space. Honestly, I chose math major without appreciating what it is but just a degree that will make me more employable in the future. We can read this as a set, say, A is stated to be a singleton/unit set if the cardinality of the set is 1 i.e. What age is too old for research advisor/professor? called the closed {\displaystyle \{S\subseteq X:x\in S\},} and A Consider $$K=\left\{ \frac 1 n \,\middle|\, n\in\mathbb N\right\}$$ } Since the complement of $\ {x\}$ is open, $\ {x\}$ is closed. By rejecting non-essential cookies, Reddit may still use certain cookies to ensure the proper functionality of our platform. Solution 3 Every singleton set is closed. I am facing difficulty in viewing what would be an open ball around a single point with a given radius? Where does this (supposedly) Gibson quote come from? { Observe that if a$\in X-{x}$ then this means that $a\neq x$ and so you can find disjoint open sets $U_1,U_2$ of $a,x$ respectively. one. {\displaystyle x} Let . Privacy Policy. Every singleton set in the real numbers is closed. {\displaystyle X} For more information, please see our A topological space is a pair, $(X,\tau)$, where $X$ is a nonempty set, and $\tau$ is a collection of subsets of $X$ such that: The elements of $\tau$ are said to be "open" (in $X$, in the topology $\tau$), and a set $C\subseteq X$ is said to be "closed" if and only if $X-C\in\tau$ (that is, if the complement is open). It depends on what topology you are looking at. "There are no points in the neighborhood of x". As has been noted, the notion of "open" and "closed" is not absolute, but depends on a topology. {\displaystyle \{y:y=x\}} S How can I see that singleton sets are closed in Hausdorff space? Honestly, I chose math major without appreciating what it is but just a degree that will make me more employable in the future. @NoahSchweber:What's wrong with chitra's answer?I think her response completely satisfied the Original post. Defn for each of their points. Well, $x\in\{x\}$. Acidity of alcohols and basicity of amines, About an argument in Famine, Affluence and Morality. The difference between the phonemes /p/ and /b/ in Japanese. I think singleton sets $\{x\}$ where $x$ is a member of $\mathbb{R}$ are both open and closed. { In the real numbers, for example, there are no isolated points; every open set is a union of open intervals. {\displaystyle X,} How many weeks of holidays does a Ph.D. student in Germany have the right to take? of x is defined to be the set B(x) Ummevery set is a subset of itself, isn't it? Is it suspicious or odd to stand by the gate of a GA airport watching the planes? Is a PhD visitor considered as a visiting scholar? is a singleton as it contains a single element (which itself is a set, however, not a singleton). vegan) just to try it, does this inconvenience the caterers and staff? . Singleton sets are not Open sets in ( R, d ) Real Analysis. For example, if a set P is neither composite nor prime, then it is a singleton set as it contains only one element i.e. There are no points in the neighborhood of $x$. a space is T1 if and only if . (6 Solutions!! How many weeks of holidays does a Ph.D. student in Germany have the right to take? But any yx is in U, since yUyU. ball, while the set {y Also, not that the particular problem asks this, but {x} is not open in the standard topology on R because it does not contain an interval as a subset. Why higher the binding energy per nucleon, more stable the nucleus is.? A singleton has the property that every function from it to any arbitrary set is injective. {\displaystyle {\hat {y}}(y=x)} N(p,r) intersection with (E-{p}) is empty equal to phi Since the complement of $\{x\}$ is open, $\{x\}$ is closed. . Euler: A baby on his lap, a cat on his back thats how he wrote his immortal works (origin?). Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Take any point a that is not in S. Let {d1,.,dn} be the set of distances |a-an|. Singleton set symbol is of the format R = {r}. Why do universities check for plagiarism in student assignments with online content? y What Is A Singleton Set? What happen if the reviewer reject, but the editor give major revision? But $y \in X -\{x\}$ implies $y\neq x$. 2 How to react to a students panic attack in an oral exam? Does ZnSO4 + H2 at high pressure reverses to Zn + H2SO4? um so? then the upward of How can I explain to my manager that a project he wishes to undertake cannot be performed by the team? Is there a proper earth ground point in this switch box? A set in maths is generally indicated by a capital letter with elements placed inside braces {}. "There are no points in the neighborhood of x". In R with usual metric, every singleton set is closed. Show that the solution vectors of a consistent nonhomoge- neous system of m linear equations in n unknowns do not form a subspace of. The cardinal number of a singleton set is one. Show that the singleton set is open in a finite metric spce. Are Singleton sets in $\mathbb{R}$ both closed and open? Is there a proper earth ground point in this switch box? There is only one possible topology on a one-point set, and it is discrete (and indiscrete). Examples: Here y takes two values -13 and +13, therefore the set is not a singleton. Does there exist an $\epsilon\gt 0$ such that $(x-\epsilon,x+\epsilon)\subseteq \{x\}$? Say X is a http://planetmath.org/node/1852T1 topological space. This is because finite intersections of the open sets will generate every set with a finite complement. is a subspace of C[a, b]. Here the subset for the set includes the null set with the set itself. $\mathbb R$ with the standard topology is connected, this means the only subsets which are both open and closed are $\phi$ and $\mathbb R$. , Thus singletone set View the full answer . Open balls in $(K, d_K)$ are easy to visualize, since they are just the open balls of $\mathbb R$ intersected with $K$. I want to know singleton sets are closed or not. Math will no longer be a tough subject, especially when you understand the concepts through visualizations. = In the real numbers, for example, there are no isolated points; every open set is a union of open intervals. For example, the set In summary, if you are talking about the usual topology on the real line, then singleton sets are closed but not open. is necessarily of this form. So $B(x, r(x)) = \{x\}$ and the latter set is open. Example 2: Find the powerset of the singleton set {5}. Every set is a subset of itself, so if that argument were valid, every set would always be "open"; but we know this is not the case in every topological space (certainly not in $\mathbb{R}$ with the "usual topology"). It depends on what topology you are looking at. The only non-singleton set with this property is the empty set. 0 and our ) In von Neumann's set-theoretic construction of the natural numbers, the number 1 is defined as the singleton Singleton will appear in the period drama as a series regular . By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. I want to know singleton sets are closed or not. X Null set is a subset of every singleton set. Doubling the cube, field extensions and minimal polynoms. Ltd.: All rights reserved, Equal Sets: Definition, Cardinality, Venn Diagram with Properties, Disjoint Set Definition, Symbol, Venn Diagram, Union with Examples, Set Difference between Two & Three Sets with Properties & Solved Examples, Polygons: Definition, Classification, Formulas with Images & Examples. The reason you give for $\{x\}$ to be open does not really make sense. Let d be the smallest of these n numbers. Every net valued in a singleton subset The singleton set is of the form A = {a}, Where A represents the set, and the small alphabet 'a' represents the element of the singleton set. X So for the standard topology on $\mathbb{R}$, singleton sets are always closed. The set is a singleton set example as there is only one element 3 whose square is 9. Metric Spaces | Lecture 47 | Every Singleton Set is a Closed Set. 1,952 . Take S to be a finite set: S= {a1,.,an}. } Example 2: Check if A = {a : a N and \(a^2 = 9\)} represents a singleton set or not? Part of solved Real Analysis questions and answers : >> Elementary Mathematics >> Real Analysis Login to Bookmark Then $x\notin (a-\epsilon,a+\epsilon)$, so $(a-\epsilon,a+\epsilon)\subseteq \mathbb{R}-\{x\}$; hence $\mathbb{R}-\{x\}$ is open, so $\{x\}$ is closed. What happen if the reviewer reject, but the editor give major revision?
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