If you are giving $\{x\}$ the subspace topology and asking whether $\{x\}$ is open in $\{x\}$ in this topology, the answer is yes. $y \in X, \ x \in cl_\underline{X}(\{y\}) \Rightarrow \forall U \in U(x): y \in U$. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. If A is any set and S is any singleton, then there exists precisely one function from A to S, the function sending every element of A to the single element of S. Thus every singleton is a terminal object in the category of sets. To show $X-\{x\}$ is open, let $y \in X -\{x\}$ be some arbitrary element. Math will no longer be a tough subject, especially when you understand the concepts through visualizations. @NoahSchweber:What's wrong with chitra's answer?I think her response completely satisfied the Original post. Acidity of alcohols and basicity of amines, About an argument in Famine, Affluence and Morality. The null set is a subset of any type of singleton set. The cardinality (i.e. Definition of closed set : My question was with the usual metric.Sorry for not mentioning that. 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. Why are physically impossible and logically impossible concepts considered separate in terms of probability? Learn more about Stack Overflow the company, and our products. I think singleton sets $\{x\}$ where $x$ is a member of $\mathbb{R}$ are both open and closed. 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. The cardinality of a singleton set is one. $y \in X, \ x \in cl_\underline{X}(\{y\}) \Rightarrow \forall U \in U(x): y \in U$, Singleton sets are closed in Hausdorff space, We've added a "Necessary cookies only" option to the cookie consent popup. 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. Find the derived set, the closure, the interior, and the boundary of each of the sets A and B. S Can I tell police to wait and call a lawyer when served with a search warrant? We walk through the proof that shows any one-point set in Hausdorff space is closed. Singleton sets are not Open sets in ( R, d ) Real Analysis. But if this is so difficult, I wonder what makes mathematicians so interested in this subject. Pi is in the closure of the rationals but is not rational. Why are trials on "Law & Order" in the New York Supreme Court? The singleton set is of the form A = {a}. rev2023.3.3.43278. n(A)=1. Redoing the align environment with a specific formatting. Proof: Let and consider the singleton set . 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. In the real numbers, for example, there are no isolated points; every open set is a union of open intervals. } Also, reach out to the test series available to examine your knowledge regarding several exams. They are also never open in the standard topology. What to do about it? It only takes a minute to sign up. so clearly {p} contains all its limit points (because phi is subset of {p}). Every net valued in a singleton subset Are Singleton sets in $\mathbb{R}$ both closed and open? = With the standard topology on R, {x} is a closed set because it is the complement of the open set (-,x) (x,). Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Anonymous sites used to attack researchers. Different proof, not requiring a complement of the singleton. Does a summoned creature play immediately after being summoned by a ready action. Thus every singleton is a terminal objectin the category of sets. called the closed I think singleton sets $\{x\}$ where $x$ is a member of $\mathbb{R}$ are both open and closed. Why do universities check for plagiarism in student assignments with online content? {\displaystyle \{0\}.}. Since the complement of $\ {x\}$ is open, $\ {x\}$ is closed. Every singleton set is closed. The set {x in R | x d } is a closed subset of C. Each singleton set {x} is a closed subset of X. Therefore, $cl_\underline{X}(\{y\}) = \{y\}$ and thus $\{y\}$ is closed. What happen if the reviewer reject, but the editor give major revision? {\displaystyle x} Suppose X is a set and Tis a collection of subsets for each of their points. Anonymous sites used to attack researchers. However, if you are considering singletons as subsets of a larger topological space, this will depend on the properties of that space. {\displaystyle \{x\}} vegan) just to try it, does this inconvenience the caterers and staff? Since a singleton set has only one element in it, it is also called a unit set. How to react to a students panic attack in an oral exam? {\displaystyle \{0\}} PhD in Mathematics, Courant Institute of Mathematical Sciences, NYU (Graduated 1987) Author has 3.1K answers and 4.3M answer views Aug 29 Since a finite union of closed sets is closed, it's enough to see that every singleton is closed, which is the same as seeing that the complement of x is open. 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. Well, $x\in\{x\}$. subset of X, and dY is the restriction is a singleton whose single element is set of limit points of {p}= phi In summary, if you are talking about the usual topology on the real line, then singleton sets are closed but not open. Every singleton is compact. Since they are disjoint, $x\not\in V$, so we have $y\in V \subseteq X-\{x\}$, proving $X -\{x\}$ is open. As has been noted, the notion of "open" and "closed" is not absolute, but depends on a topology. 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. The proposition is subsequently used to define the cardinal number 1 as, That is, 1 is the class of singletons. Is a PhD visitor considered as a visiting scholar? What happen if the reviewer reject, but the editor give major revision? ) So $r(x) > 0$. Open Set||Theorem of open set||Every set of topological space is open IFF each singleton set open . The two subsets are the null set, and the singleton set itself. Structures built on singletons often serve as terminal objects or zero objects of various categories: Let S be a class defined by an indicator function, The following definition was introduced by Whitehead and Russell[3], The symbol 18. 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 . 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 $(x-\epsilon,x+\epsilon)\subseteq\mathcal{O}$. { Every nite point set in a Hausdor space X is closed. How much solvent do you add for a 1:20 dilution, and why is it called 1 to 20? Reddit and its partners use cookies and similar technologies to provide you with a better experience. ^ Singleton Set has only one element in them. { The cardinal number of a singleton set is one. y Each of the following is an example of a closed set. , For a set A = {a}, the two subsets are { }, and {a}. 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 $\epsilon=|a-x|$. } The following result introduces a new separation axiom. {\displaystyle \iota } Prove Theorem 4.2. The powerset of a singleton set has a cardinal number of 2. Metric Spaces | Lecture 47 | Every Singleton Set is a Closed Set. in a metric space is an open set. The difference between the phonemes /p/ and /b/ in Japanese. (Calculus required) Show that the set of continuous functions on [a, b] such that. Sets in mathematics and set theory are a well-described grouping of objects/letters/numbers/ elements/shapes, etc. If you preorder a special airline meal (e.g. Why higher the binding energy per nucleon, more stable the nucleus is.? 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. Now lets say we have a topological space X in which {x} is closed for every xX. 690 07 : 41. x In von Neumann's set-theoretic construction of the natural numbers, the number 1 is defined as the singleton Example 1: Which of the following is a singleton set? Since all the complements are open too, every set is also closed. Why do universities check for plagiarism in student assignments with online content? A subset C of a metric space X is called closed 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. y Assume for a Topological space $(X,\mathcal{T})$ that the singleton sets $\{x\} \subset X$ are closed. This states that there are two subsets for the set R and they are empty set + set itself. Each open -neighborhood Inverse image of singleton sets under continuous map between compact Hausdorff topological spaces, Confusion about subsets of Hausdorff spaces being closed or open, Irreducible mapping between compact Hausdorff spaces with no singleton fibers, Singleton subset of Hausdorff set $S$ with discrete topology $\mathcal T$. Lemma 1: Let be a metric space. . and our It is enough to prove that the complement is open. 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. Null set is a subset of every singleton set. The power set can be formed by taking these subsets as it elements. 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. Already have an account? What to do about it? Expert Answer. That takes care of that. A set in maths is generally indicated by a capital letter with elements placed inside braces {}. bluesam3 2 yr. ago The singleton set has only one element in it. By rejecting non-essential cookies, Reddit may still use certain cookies to ensure the proper functionality of our platform. := {y for X. Ranjan Khatu. Part of solved Real Analysis questions and answers : >> Elementary Mathematics >> Real Analysis Login to Bookmark Does ZnSO4 + H2 at high pressure reverses to Zn + H2SO4? Well, $x\in\{x\}$. { How much solvent do you add for a 1:20 dilution, and why is it called 1 to 20? Quadrilateral: Learn Definition, Types, Formula, Perimeter, Area, Sides, Angles using Examples! The reason you give for $\{x\}$ to be open does not really make sense. Lets show that {x} is closed for every xX: The T1 axiom (http://planetmath.org/T1Space) gives us, for every y distinct from x, an open Uy that contains y but not x. a space is T1 if and only if . Honestly, I chose math major without appreciating what it is but just a degree that will make me more employable in the future. {\displaystyle X.} Follow Up: struct sockaddr storage initialization by network format-string, Acidity of alcohols and basicity of amines. $U$ and $V$ are disjoint non-empty open sets in a Hausdorff space $X$. is a subspace of C[a, b]. In mathematics, a singleton, also known as a unit set[1] or one-point set, is a set with exactly one element. x. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. of X with the properties. Whole numbers less than 2 are 1 and 0. } Are these subsets open, closed, both or neither? ), Are singleton set both open or closed | topology induced by metric, Lecture 3 | Collection of singletons generate discrete topology | Topology by James R Munkres. } How many weeks of holidays does a Ph.D. student in Germany have the right to take? "There are no points in the neighborhood of x". PS. A subset O of X is {\displaystyle X,} Solution:Given set is A = {a : a N and \(a^2 = 9\)}. Then $X\setminus \{x\} = (-\infty, x)\cup(x,\infty)$ which is the union of two open sets, hence open. Consider $\{x\}$ in $\mathbb{R}$. and It is enough to prove that the complement is open. ncdu: What's going on with this second size column? So: is $\{x\}$ open in $\mathbb{R}$ in the usual topology? X . Here $U(x)$ is a neighbourhood filter of the point $x$. But $y \in X -\{x\}$ implies $y\neq x$. Use Theorem 4.2 to show that the vectors , , and the vectors , span the same . There are no points in the neighborhood of $x$. Equivalently, finite unions of the closed sets will generate every finite set. They are also never open in the standard topology. is called a topological space Show that the singleton set is open in a finite metric spce. which is the set aka equipped with the standard metric $d_K(x,y) = |x-y|$. 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"). so, set {p} has no limit points , Thus singletone set View the full answer . In particular, singletons form closed sets in a Hausdor space. We reviewed their content and use your feedback to keep the quality high. At the n-th . } Connect and share knowledge within a single location that is structured and easy to search. Summing up the article; a singleton set includes only one element with two subsets. Ummevery set is a subset of itself, isn't it? X A set is a singleton if and only if its cardinality is 1. A In the given format R = {r}; R is the set and r denotes the element of the set. All sets are subsets of themselves. : NOTE:This fact is not true for arbitrary topological spaces. The following topics help in a better understanding of singleton set. if its complement is open in X. 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 }. Defn 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. If there is no such $\epsilon$, and you prove that, then congratulations, you have shown that $\{x\}$ is not open. The set {y Let E be a subset of metric space (x,d). Who are the experts? {\displaystyle x} 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 . A singleton set is a set containing only one element. Arbitrary intersectons of open sets need not be open: Defn empty set, finite set, singleton set, equal set, disjoint set, equivalent set, subsets, power set, universal set, superset, and infinite set. The given set has 5 elements and it has 5 subsets which can have only one element and are singleton sets. Then every punctured set $X/\{x\}$ is open in this topology. Um, yes there are $(x - \epsilon, x + \epsilon)$ have points. When $\{x\}$ is open in a space $X$, then $x$ is called an isolated point of $X$. 3 As Trevor indicates, the condition that points are closed is (equivalent to) the $T_1$ condition, and in particular is true in every metric space, including $\mathbb{R}$. We are quite clear with the definition now, next in line is the notation of the set. 0 The only non-singleton set with this property is the empty set. Then the set a-d<x<a+d is also in the complement of S. Example: Consider a set A that holds whole numbers that are not natural numbers. X in X | d(x,y) }is $\mathbb R$ with the standard topology is connected, this means the only subsets which are both open and closed are $\phi$ and $\mathbb R$. However, if you are considering singletons as subsets of a larger topological space, this will depend on the properties of that space. Since the complement of $\{x\}$ is open, $\{x\}$ is closed. Then for each the singleton set is closed in . The singleton set has two subsets, which is the null set, and the set itself. So $B(x, r(x)) = \{x\}$ and the latter set is open. Here's one. is necessarily of this form. So that argument certainly does not work. The two subsets of a singleton set are the null set, and the singleton set itself. {\displaystyle X} A singleton set is a set containing only one element. 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 $X\setminus \{x\}.$. X How many weeks of holidays does a Ph.D. student in Germany have the right to take? {\displaystyle x\in X} Why do many companies reject expired SSL certificates as bugs in bug bounties? In general "how do you prove" is when you . is a singleton as it contains a single element (which itself is a set, however, not a singleton). To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Also, the cardinality for such a type of set is one. x {y} { y } is closed by hypothesis, so its complement is open, and our search is over. Share Cite Follow answered May 18, 2020 at 4:47 Wlod AA 2,069 6 10 Add a comment 0 This is what I did: every finite metric space is a discrete space and hence every singleton set is open. ball, while the set {y x the closure of the set of even integers. i.e. By the Hausdorff property, there are open, disjoint $U,V$ so that $x \in U$ and $y\in V$. The set A = {a, e, i , o, u}, has 5 elements. Show that the singleton set is open in a finite metric spce. Title. Therefore the powerset of the singleton set A is {{ }, {5}}. Do I need a thermal expansion tank if I already have a pressure tank? "Singleton sets are open because {x} is a subset of itself. " Check out this article on Complement of a Set. 1,952 . What Is the Difference Between 'Man' And 'Son of Man' in Num 23:19? Ranjan Khatu. {\displaystyle \{\{1,2,3\}\}} Examples: 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). in Tis called a neighborhood Metric Spaces | Lecture 47 | Every Singleton Set is a Closed Set, Singleton sets are not Open sets in ( R, d ), 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, Theorem: Every subset of topological space is open iff each singleton set is open. So: is $\{x\}$ open in $\mathbb{R}$ in the usual topology? Since were in a topological space, we can take the union of all these open sets to get a new open set. What age is too old for research advisor/professor? Can I take the open ball around an natural number $n$ with radius $\frac{1}{2n(n+1)}$?? Why does [Ni(gly)2] show optical isomerism despite having no chiral carbon? Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. 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. "There are no points in the neighborhood of x". That is, why is $X\setminus \{x\}$ open? { , A singleton has the property that every function from it to any arbitrary set is injective. { Then $(K,d_K)$ is isometric to your space $(\mathbb N, d)$ via $\mathbb N\to K, n\mapsto \frac 1 n$. This is a minimum of finitely many strictly positive numbers (as all $d(x,y) > 0$ when $x \neq y$). I want to know singleton sets are closed or not. X 968 06 : 46. How can I see that singleton sets are closed in Hausdorff space? 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). 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. Defn 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. in You can also set lines='auto' to auto-detect whether the JSON file is newline-delimited.. Other JSON Formats. Take any point a that is not in S. Let {d1,.,dn} be the set of distances |a-an|. 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 $x\in B\subset A$). Share Cite Follow edited Mar 25, 2015 at 5:20 user147263 So in order to answer your question one must first ask what topology you are considering. Connect and share knowledge within a single location that is structured and easy to search. 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. metric-spaces. { one. It depends on what topology you are looking at. Every singleton set in the real numbers is closed. If there is no such $\epsilon$, and you prove that, then congratulations, you have shown that $\{x\}$ is not open. The singleton set has only one element, and hence a singleton set is also called a unit set. X , x one. {x} is the complement of U, closed because U is open: None of the Uy contain x, so U doesnt contain x. Metric Spaces | Lecture 47 | Every Singleton Set is a Closed Set, Singleton sets are not Open sets in ( R, d ), Are Singleton sets in $mathbb{R}$ both closed and open? then (X, T) What does that have to do with being open? Every singleton set is an ultra prefilter. X is a principal ultrafilter on If Privacy Policy. } As the number of elements is two in these sets therefore the number of subsets is two. Examples: Example 3: Check if Y= {y: |y|=13 and y Z} is a singleton set? x I am afraid I am not smart enough to have chosen this major. { 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}. Every singleton set is an ultra prefilter. When $\{x\}$ is open in a space $X$, then $x$ is called an isolated point of $X$. 690 14 : 18. Take S to be a finite set: S= {a1,.,an}. denotes the singleton {\displaystyle 0} The subsets are the null set and the set itself. The number of subsets of a singleton set is two, which is the empty set and the set itself with the single element. What happen if the reviewer reject, but the editor give major revision? Say X is a http://planetmath.org/node/1852T1 topological space. What is the correct way to screw wall and ceiling drywalls? Metric Spaces | Lecture 47 | Every Singleton Set is a Closed Set. Notice that, by Theorem 17.8, Hausdor spaces satisfy the new condition. Why do small African island nations perform better than African continental nations, considering democracy and human development? I am facing difficulty in viewing what would be an open ball around a single point with a given radius? y For every point $a$ distinct from $x$, there is an open set containing $a$ that does not contain $x$. Example 2: Check if A = {a : a N and \(a^2 = 9\)} represents a singleton set or not? What video game is Charlie playing in Poker Face S01E07? Moreover, each O Has 90% of ice around Antarctica disappeared in less than a decade? Having learned about the meaning and notation, let us foot towards some solved examples for the same, to use the above concepts mathematically. In the real numbers, for example, there are no isolated points; every open set is a union of open intervals. Then $X\setminus \ {x\} = (-\infty, x)\cup (x,\infty)$ which is the union of two open sets, hence open. Doubling the cube, field extensions and minimal polynoms. Therefore the five singleton sets which are subsets of the given set A is {1}, {3}, {5}, {7}, {11}. This topology is what is called the "usual" (or "metric") topology on $\mathbb{R}$. "Singleton sets are open because {x} is a subset of itself. " So for the standard topology on $\mathbb{R}$, singleton sets are always closed. which is the same as the singleton E is said to be closed if E contains all its limit points. Why does [Ni(gly)2] show optical isomerism despite having no chiral carbon? Honestly, I chose math major without appreciating what it is but just a degree that will make me more employable in the future. Solution 4. (6 Solutions!! Where does this (supposedly) Gibson quote come from? This is definition 52.01 (p.363 ibid. . 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). Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. A set such as 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 $\epsilon=|a-x|$. 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. This is because finite intersections of the open sets will generate every set with a finite complement. Learn more about Stack Overflow the company, and our products. Why higher the binding energy per nucleon, more stable the nucleus is.? The idea is to show that complement of a singleton is open, which is nea. The complement of is which we want to prove is an open set. 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 This is because finite intersections of the open sets will generate every set with a finite complement. But if this is so difficult, I wonder what makes mathematicians so interested in this subject. Note. {\displaystyle {\hat {y}}(y=x)} ), 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. {y} is closed by hypothesis, so its complement is open, and our search is over. Let . Let us learn more about the properties of singleton set, with examples, FAQs. {\displaystyle \{A,A\},} S in X | d(x,y) = }is Hence $U_1$ $\cap$ $\{$ x $\}$ is empty which means that $U_1$ is contained in the complement of the singleton set consisting of the element x. What are subsets of $\mathbb{R}$ with standard topology such that they are both open and closed? is a set and and Tis called a topology
show that every singleton set is a closed set