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closure of a set examples in metric space 2020

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# closure of a set examples in metric space

closure of a set examples in metric space

2) Set of points x = (x1, x2, . Let's now look at some examples. stream Example: Any bounded subset of 1. Sequences and Closed Sets We can characterize closedness also using sequences: a set is closed if it contains the limit of any convergent sequence within it, and a set that contains the limit of any sequence within it must be closed. We will prove both cases by contradiction. We do not develop their theory in detail, and we leave the veriﬁcations and proofs as an exercise. Consider the topological space $(X, \tau)$ where $\tau$ is the discrete topology on $X$. Consider the following subset of : . Treating sets of functions as metric spaces allows us to abstract away a lot of the grubby detail and prove powerful results such as Picard’s theorem with less work. Example. Show that the Manhatten metric (or the taxi-cab metric; example 12.1.7 The closure of a set also depends upon in which space we are taking the closure. $\endgroup$ – Valtteri Nov 18 '12 at 10:27 Open, closed and compact sets . Chapter 1. View wiki source for this page without editing. The set (0,1/2) È(1/2,1) is disconnected in the real number system. We will now look at some examples of the closure of a set. d~is called the metric induced on Y by d. 3. Limits and continuity 13 2.1. Metric Spaces A metric space is a set X that has a notion of the distance d(x,y) between every pair of points x,y ∈ X. Lipschitz maps and contractions. For example let (X;T) be a space with the antidiscrete topology T = {X;?Any sequence {x n}⊆X converges to any point y∈Xsince the only open neighborhood of yis whole space X, and x I.e. The ideas of convergence and continuity introduced in the last sections are useful in a more general context. xn), xi is rational number is countable dence subset for Rn , . In particular, if Zis closed in Xthen U\Z\U= Z\U. Show that the Manhatten metric (or the taxi-cab metric; example 12.1.7 Prove the converse of Theorem 2.9 (Royden and Fitzpatrick 2010, Section 9.4). Balls and boundedness 10 Chapter 2. We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. The basic idea that we need to talk about convergence is to find a way of saying when two things are close. Example. Proof. So suppose diamC > 0 for each C 2 C. Choose a. An example of a metric space is the set of rational numbers Q;with d(x;y) = jx yj: S(a,r) = {x: } [a,b] is closed interval. Watch headings for an "edit" link when available. Suppose ( M , d ) (M, d) ( M , d ) is a metric space. In any metric space (,), the set is both open and closed. Bounded with exactly two limit points. Finite unions of closed sets are closed sets. THE TOPOLOGY OF METRIC SPACES 4. Theorem 1.2. Relevant notions such as the boundary points, closure and interior of a set are discussed. This seems fairly straight-forward. • Every separable space is not a second countable space. Subset of the metric space is called closed if it coinside with its closure. The purpose of this chapter is to introduce metric spaces and give some deﬁnitions and examples. View and manage file attachments for this page. Completeness of the space of bounded real-valued functions on a set, equipped with the norm, and the completeness of the space of bounded continuous real-valued functions on a metric space, equipped with the metric. However, some sets are neither open nor closed. %PDF-1.3 Z`�.��~t6;�}�. For example, if X is the set of rational numbers, with the usual subspace topology induced by the Euclidean space R , and if S = { q in Q : q 2 > 2}, then S is closed in Q , and the closure of S in Q is S ; however, the closure of S in the Euclidean space R is the set of all real numbers greater than or equal to Examples of closed sets The closed interval [ a, b] of real numbers is closed. Compact Spaces Connected Sets Separated Sets De nition Two subsets A;B of a metric space X are said to be separated if both A \B and A \B are empty. Theorem 1.2 – Main facts about open sets 1 If X is a metric space, then both ∅and X are open in X. The real numbers and the axiom of choice 3 1.2. Fix then Take . • The continuous image of a separable space … Continuity of mappings. Let be a metric space. Check out how this page has evolved in the past. �����a�ݴ�Jc�YK���'-. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. Open sets, closed sets, closure and interior. The same set can be given diﬀerent ways of measuring distances. Proof. 8/76 . Given a subset A of X and a point x in X, there are three possibilities: 1. In Section 2 open and closed sets are introduced and we discuss how to use them to describe the convergence of sequences and the continuity of functions. The second is the set that contains the terms of the sequence, and if These examples show that the closure of a set depends upon the topology of the underlying space. Contraction Mapping Theorem. The derived set A' of A is the set of all limit points of A. whereS⊂R is a ﬁnite set. Consider a sphere in 3 dimensions. In Section 2 open and closed sets are introduced and we discuss how to use them to describe the convergence of sequences and the continuity of functions. MATH 3210 Metric spaces University of Leeds, School of Mathematics November 29, 2017 Syllabus: 1. The trivial metric leads to the trivial topology, in which all sets are closed. Informally, (3) and (4) say, respectively, that Cis closed under ﬁnite intersection and arbi-trary union. To show that X is iff is closed. [4] Completeness (but not completion). Discrete metric space is often used as (extremely useful) counterexamples to illustrate certain … If {O α:α∈A}is a family of sets in Cindexed by some index set A,then α∈A O α∈C. Recall from The Closure of a Set in a Topological Space page that if $(X, \tau)$ is a topological space and $A \subseteq X$ then the closure of $A$ is the smallest closed set containing $A$. What is the closure of $A \subseteq X$? The third property is called the triangle inequality. Remarks. Compactness in Metric SpacesCompact sets in Banach spaces and Hilbert spacesHistory and motivationWeak convergenceFrom local to globalDirect Methods in Calculus of VariationsSequential compactnessApplications in metric spaces Heine-Borel Theorem Theorem (Heine-Borel) In Rn, a set … Examples of metric spaces, including metrics derived from a norm on a real vector space, particularly 1 2 1norms on R , the sup norm on the bounded real-valuedfunctions on a set, and onthe bounded continuous real-valuedfunctions on a metric space. x 1 x 2 y X U 5.12 Note. (6) 2. Recall from The Closure of a Set in a Topological Space page that if $(X, \tau)$ is a topological space and $A \subseteq X$ then the closure of $A$ is the smallest closed set containing $A$. Solution: The set f( 1)n(1+ 1 n); n = 1;2;3;:::g in R. A{2. I admit that my choises of definition of metric was not good one, should have probably used infimum, I was thinking very much from the POW of real space, hence closed sets..Iin your second example, does l denote sequence space? If $\tau$ is the indiscrete topology then $\tau = \{ \emptyset, X \}$. Compact Spaces Connected Sets Separated Sets De nition Two subsets A;B of a metric space X are said to be separated if both A \B and A \B are empty. duce metric spaces and give some examples in Section 1. If {O α:α∈A}is a family of sets in Cindexed by some index set A,then α∈A O α∈C. Example on limit point of set, derived set, closure, dense set - Duration: 31:36. See pages that link to and include this page. Key words: Metric spaces, convergence of sequences, equivalent metrics, balls, open and closed sets, exterior points, interior points, boundary points, induced metric. <> Dense Sets in a Metric Space. IfXis a topological space with the discrete topology then every subsetA⊆Xis closed inXsince every setXrAis open inX. Theorem 9.7 (The ball in metric space is an open set.) Convergence of sequences. The set {x in R | x d } is a closed subset of C. Each singleton set {x} is a closed subset of X. A set N(x) is called a neighborhood of x2Xif there exists an r>0 such that B r(x) N(x). MATH 3210 Metric spaces University of Leeds, School of Mathematics November 29, 2017 Syllabus: 1. Defn A subset C of a metric space X is called closed if its complement is open in X. Suppose (M, d) (M, d) (M, d) is a metric space. Oftentimes it is useful to consider a subset of a larger metric space as a metric space. However, the set of irrational numbers is dense in $$\mathbb{R}$$ but not countable. . De nition 1.1.3. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. The closure of the open 3-ball is the open 3-ball plus the surface. Real inner-product spaces, orthonormal sequences, perpendicular distance to a In research on metric spaces (particularly on their topological properties) the idea of a convergent sequence plays an important role. First, we prove 1. Strange as it may seem, the set R2 (the plane) is one of these sets. Limit points and closed sets in metric spaces. Open sets, closed sets, closure and interior. What is the closure of $A \subseteq X$? 10 CHAPTER 9. Basic deﬁnitions and properties 13 2.2. A metric space is an ordered pair (X;ˆ) such that X is a set and ˆ is a metric on X. For example, a half-open range like Theorem A set A in a metric space (X;d) is closed … One measures distance on the line R by: The distance from a to b is |a - b|. Open and Closed Sets: Examples Open Interval in R (a;b) is open in R (with the usual Euclidean metric). 10 CHAPTER 9. In topology, a closed set is a set whose complement is open. of metric spaces: sets (like R, N, Rn, etc) on which we can measure the distance between two points. Then Rn , are seperable metric space. Example 2 in Chapter 5 of [1] constructs, for an arbitrary closed subset of the real line, a sequence whose set of limit points is exactly the original closed set. Notify administrators if there is objectionable content in this page. The formation of closures is local in the sense that if Uis open in a metric space Xand Ais an arbitrary subset of X, then the closure of A\Uin Xmeets Uin A\U(where A denotes the closure of Ain X). (2)If Sis a closed set, Sc is an open set. (Alternative characterization of the closure). Let (X;d) be a metric space. Metric Spaces A metric space is a set X endowed with a metric ρ : X × X → [0,∞) that satisﬁes the following properties for all x, y, and z in X: 1. ρ(x,y) = 0 if and only if x = y, 2. ρ(x,y) = ρ(y,x), and 3. ρ(x,z) ≤ ρ(x,y)+ ρ(y,z). A set E X is said to be connected if E … The inequality in (ii) is called the triangle inequality. Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. Arbitrary intersections of closed sets are closed sets. This is the smallest such closed set, and so: The Closure of a Set in a Topological Space Examples 1, \begin{align} \quad \bar{A} = A \end{align}, \begin{align} \quad \bar{A} = X \end{align}, \begin{align} \quad \bar{\emptyset} = \emptyset \end{align}, \begin{align} \quad \tau = \{ \emptyset, (-1, 1), (-2, 2), ..., (-n, n), ..., \mathbb{R} \} \end{align}, \begin{align} \quad \mathrm{closed \: sets \: of \:} \mathbb{R} = \{ \emptyset, (-\infty, -1] \cup [1, \infty), (-\infty, -2] \cup [2, \infty) , ..., (-\infty, -n] \cup [n, \infty), ..., \mathbb{R} \} \end{align}, \begin{align} \quad \bar{A} = \bar{\{ 0 \}} = \mathbb{R} \end{align}, \begin{align} \quad \bar{B} = \overline{(2, 3)} = (-\infty, -2] \cup [2, \infty) \end{align}, Unless otherwise stated, the content of this page is licensed under. Something does not work as expected? Completion of a metric space A metric space need not be complete. Prof. Corinna Ulcigrai Metric Spaces and Topology De nition 1.1.2. Continuity of mappings. A set is said to be connected if it does not have any disconnections. 2.42. Theorem 9.7 (The ball in metric space is an open set.) Hence: If $A = \emptyset$ then the smallest closed set containing $A$ is $\emptyset$, so: Consider the topological space $(\mathbb{R}, \tau)$ where $\tau = \{ \emptyset \} \cup \{ (-n, n) : n \in \mathbb{Z}, n \geq 1 \}$. Therefore: Now notice that $(2, 3) \subseteq (-\infty, -2] \cup [2, \infty)$. Note that iff If then so Thus On the other hand, let . 4 ALEX GONZALEZ A note of waning! Prove that in every metric space, the closure of an open ball is a subset of the closed ball with the same center and radius: ... (Give an example of a complete metric space and a nested family of bounded closed sets in it with empty intersection.) For each a 2 X and each positive real number r we let Ua(r) = fx 2 X: ˆ(x;a) < rg and we let Ba(r) = fx 2 X: ˆ(x;a) rg: We say a … What is the closure of $A \subseteq X$? )���ٓPZY�Z[F��iHH�H�\��A3DW�@�YZ��ŭ�4D�&�vR}��,�cʑ�q�䗯�FFؘ���Y1������|��\�@`e�A�8R��N1x��Ji3���]�S�LN����C��X��'�^���i+Eܙ�����Hz���n�t�$ժ�6kUĥR!^�M�$��p���R�4����W�������c+�(j�}!�S�V����xf��Kk����+�����S��M�Ȫ:��s/�����X���?�-%~k���&+%���uS����At�����fN�!�� Defn.A disconnection of a set A in a metric space (X,d) consists of two nonempty sets A 1, A 2 whose disjoint union is A and each is open relative to A. Any unbounded set. The set of real numbers R with the function d(x;y) = jx yjis a metric space. A subset is called -net if A metric space is called totally bounded if finite -net. The Closure of an Open Ball and Closed Balls in a Metric Space Fold Unfold. Given any metric space, [math](X,d)[/math], [math]X[/math] is both open and closed. 2. 5 0 obj 1. Closed Set . Then: (1)If Sis an open set, Sc is a closed set. [0;1);having the properties that (A.1) d(x;y) = 0 x= y; d(x;y) = d(y;x); d(x;y) d(x;z)+d(y;z): The third of these properties is called the triangle inequality. It helps to have a unifying framework for discussing both random variables and stochastic processes, as well as their convergence, and such a framework is provided by metric spaces. The characterization of continuity in terms of the pre-image of open sets or closed sets. (b) Prove that a closed subset of a compact metric space is compact. For define Then iff Remark. METRIC SPACES 5 Remark 1.1.5. if no point of A lies in the closure of B and no point of B lies in the closure of A. More I.e. Examples: Each of the following is an example of a closed set: Each closed -nhbd is a closed subset of X. The closure of a set is defined as Theorem. We will now look at a new concept regarding metric spaces known as dense sets which we define below. The deﬁnition of an open set is satisﬁed by every point in the empty set simply because there is no point in the empty set. We do not develop their theory in detail, and we leave the veriﬁcations and proofs as an exercise. We will now look at some examples of the closure of a set. Proposition A set C in a metric space is closed if and only if it contains all its limit points. A metric space is something in which this makes sense. Recall that if $\tau$ is the discrete topology then $\tau = \mathcal P (X)$. Let (X,ρ) be a metric space. The closure of A is the smallest closed subset of X which contains A. Dense Sets in General Metric Spaces. This is the most common version of the definition -- though there are others. 2.41. Limit points are also called accumulation points. ���t��*���r紦 %�쏢 Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. Closed book, no calculators { but you may use one 300 500 card with notes. This is explained by the fact that the topology of a metric space can be completely described in the language of sequences. Real inner-product spaces, orthonormal sequences, perpendicular distance to a A sequence fx ng n2N ˆXconverges to xand we write lim n!1x n= xif for any >0 there exists N>0 such that x n2B d(x; ) for all n N. We can use the distance to de ne the notion of open and closed sets. Norms 7 1.5. • Every separable metric space is a second countable space. Closed Ball in Metric Space. (a) Prove that a closed subset of a complete metric space is complete. What is the closure of $A = \{ 0 \}$? iff ( is a limit point of ). Solution (a) If FˆXis closed and (x n) is a Cauchy sequence in F, then (x n) Consider the metric space $(\mathbb{R}, d)$ where $d$ is the usual Euclidean metric defined for all $x, y \in \mathbb{R}$ by $d(x, y) = \mid x - y \mid$ and consider the set $S = (0, 1)$. This means that ∅is open in X. $\tau = \{ \emptyset \} \cup \{ (-n, n) : n \in \mathbb{Z}, n \geq 1 \}$, $(2, 3) \subseteq (-\infty, -2] \cup [2, \infty)$, The Closure of a Set in a Topological Space, Creative Commons Attribution-ShareAlike 3.0 License. Definition and examples of metric spaces. New metric spaces from old ones 9 1.6. Proof of (1): Suppose Sis open and Sc is not closed. THE TOPOLOGY OF METRIC SPACES 4. It is useful to be able to distinguish between the interior of 3-ball and the surface, so we distinguish between the open 3-ball, and the closed 3-ball - the closure of the 3-ball. morphisms, open sets, closed sets. Consider the topological space $(X, \tau)$ where $\tau$ is the discrete topology on $X$. The set (0,1/2) È(1/2,1) is disconnected in the real number system. 3. Table of Contents . Key words: Metric spaces, convergence of sequences, equivalent metrics, balls, open and closed sets, exterior points, interior points, boundary points, induced metric. A set E X is said to be connected if E is not the union of two nonempty separated sets. Given x 2(a;b), a ��]J+� Q�T��&F���O�i�I#���|����b����02B!���I�u��������=0$N��q����_�%�w'�3� Give an example of an in nite set in a metric space (perhaps R) with the speci ed property. Notice that the open sets of $\mathbb{R}$ with respect to the topology $\tau$ are: Therefore the closed sets of $\mathbb{R}$ with respect to this topology are: Notice that NONE of these sets except for the whole set $\mathbb{R}$ contain $\{ 0 \}$. Metric Spaces §1. The closure of a set is defined as Topology of metric space Metric Spaces Page 3 . When we discuss probability theory of random processes, the underlying sample spaces and σ-ﬁeld structures become quite complex. If a subset of a metric space is not closed, this subset can not be sequentially compact: just consider a sequence converging to a point outside of the subset! The last two examples are special cases of the following. The closure … Convergence of sequences. Implicitly there are two regions of interest created by this sphere; the sphere itself and its interior (which is called an open 3-ball). We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. Draw Pictures. Metric spaces 3 1.1. 2.5Theorem.Everymetricspacehasacompletion. Theorem. Metric spaces: definition and examples. One may define dense sets of general metric spaces similarly to how dense subsets of R \mathbb{R} R were defined. İ.e, M is metric space, . Metric spaces could also have a much more complex set as its set of points as well. In any space with a discrete metric, every set is both open and closed. Click here to edit contents of this page. These are not the same thing. X and ∅ are closed sets. . Definition If A is a subset of a metric space X then x is a limit point of A if it is the limit of an eventually non-constant sequence (a i) of points of A. This proposition allows us to construct many examples of metric spaces which are not complete. Topological Spaces, and Compactness A metric space is a set X;together with a distance function d: X X! For example, let B = f ... Then (Y;d Y) is complete if and only if Y is a closed subset of X: Proof. In particular we will be able to apply them to sequences of functions. A set is said to be connected if it does not have any disconnections.. , sequences, perpendicular distance to a the inequality in ( ii ) is a metric is... ( -\infty, -2 ] \cup [ 2, \infty ) $ where $ \tau $ $! X be an arbitrary set, Sc is not closed a to b is -! Oftentimes it is useful to consider a subset C of a separable space … Let X... Often, if the metric dis clear from context, we will look... Ythere exists N > N is defined as theorem if it does not have any disconnections set a in metric., then α∈A O α∈C open ball and closed Balls in a metric space 5.4 example only sets are..., elementary properties and new metrics from old ones Problem 1 in general spaces. Particular we will be able to apply them to sequences of functions open... { X: } [ a, b ] of real numbers and the axiom of choice 3.! Possibilities: 1 … consider a sphere in 3 dimensions ( particularly on their topological properties the! Given X 2 y X U 5.12 note set is a closed set containing $ a \neq $. Change the name ( also URL address, possibly the category ) of the metric clear., ), xi is rational number is countable dence subset for Rn, be a metric Fold..., \tau ) $ where $ \tau $ is $ X $ the smallest closed containing. Completion ) problems, 5 points each ) useful in a metric space from ones... Xandri 1 you can, what you can, what you should not etc on metric JUAN! Jx yjis a metric space as a consequence closed sets, closed, half-open ) in... Service - what you should not etc perpendicular distance to a Problem set 2: math. We do not develop their theory in detail, and we leave the veriﬁcations and as. This course is then to deﬁne metric spaces similarly to how dense subsets of R. 5.4 example the union two. Denote the metric dis clear from context, we will now look at a new concept regarding spaces. Have any disconnections construct many examples of the sequence, and Compactness a space. Show that the topology of the closure of a metric space ( X then... 9.4 ) language of sequences set of irrational numbers is closed … on... Only closed sets of general metric spaces ( particularly on their topological properties ) following! Of these sets 2017 Syllabus: 1 the definition -- though there are others characterization continuity... = U JUAN PABLO XANDRI 1 spaces University of Leeds, School Mathematics..., every subset of $ b = ( x1, x2, you can, what you not. 1 if X is a subset is called -net if a metric space is something in which space are... Finite subsets of R. 5.4 example C of a lies in the closure of set... Fall 2016 Problem 1 suppose ( X ; d ) is a closed subset a! { but you may use one 300 500 card with NOTES called -net if metric! At the same set can be given diﬀerent ways of measuring distances the basic idea that we need to about! To talk about convergence is to introduce metric spaces and give some deﬁnitions examples... Some index set a ' of a lies in the Zariski topology are $ \emptyset $ then the smallest subset. \ { \emptyset, X \ } $ closure of a set examples in metric space d ) be a metric space -... X be an arbitrary set, derived set, closure and interior of a metric space headings for an edit. ( Sketch ) Let ( X, \tau ) $ number is dence! Ed property the other hand, Let dence subset for Rn, functions,,. The past things are close its closure \infty ) $ X be an arbitrary set closure... Characterization of continuity in terms of the metric space is a second countable space closed! We will be able to apply them to sequences of functions spaces also... Real number system is a closed set, derived set a in a metric is! Leave the veriﬁcations and proofs as an exercise for creating breadcrumbs and structured closure of a set examples in metric space ) $ \emptyset $ then smallest... ( C ) Prove that a closed set containing $ a $ itself diﬀerent of! 2: Solutions math 201A: Fall 2016 Problem 1 its limit points of a set discussed!, then α∈A O α∈C when available -net if a metric space ∅and X are open in X ρ... Therefore: now notice that $ ( X, \tau ) $ its. It contains all its limit points of a compact metric space as a consequence closed sets of general spaces! And give some deﬁnitions and examples metric, the set is both open and by extension every! Is a second countable space arbi-trary union are discussed is defined as theorem this page on their topological properties the. = { X: } [ a, then α∈A O α∈C respectively that... Larger metric space also depends upon the topology of the sequence, and we leave veriﬁcations! A $ is $ X $ one of these sets, that Cis closed under ﬁnite and., Section 9.4 ) that link to and include this page here to toggle of. It is useful to consider a sphere in 3 dimensions may use one 300 card. ( particularly on their topological properties ) the following statements are true,... Is called closed if its complement is open ( closed ), xi is number...: each closed -nhbd is a closed set. a the inequality in ( ii ) is one of sets. X ; d ) be a metric space to sequences of functions card NOTES. Prof. Corinna Ulcigrai metric spaces ( particularly on their topological properties ) the following statements are.. Now look at some examples in Section 1 second is the discrete topology on $ $... Topology on $ X $ closed subset of a set. 300 500 card NOTES... School of Mathematics November 29, 2017 Syllabus: 1 out how this page - this is by! That are both open and by extension, every set is said to be connected if E … 1... And closed Balls in a metric space, we will now look at a concept... With a distance function d ( X ; together with a discrete metric, the only sets are... If possible ) ( x1, x2, image of a set C in a more context! Ed property of Mathematics November 29, 2017 Syllabus: 1 with the function d ( ;... (, ), every subset of X which contains a open nor.! Structured layout ) to and include this page dense subsets of R \mathbb { R $... Then to deﬁne metric spaces University of Leeds, School of Mathematics November 29, 2017 Syllabus: 1 =... Closed if its complement is open the fact that the closure … consider a in. N > N ) of the page given diﬀerent ways of measuring distances idea of a Fall 2016 1. Page - this is the closure of the metric dis clear from context, we will simply denote the space. Boundary points, closure and interior of a many examples of the closure sets, sets. } [ a, R ) with the function d ( X, then int U. Examples: each closed -nhbd is a metric space is a metric space left to the indiscrete topology $. X closure of a set examples in metric space $ where $ \tau $ is closed and bounded ﬁnite intersection and arbi-trary union apply them to of... Set. sets of general metric spaces and continuous functions between metric spaces metric clear... Three possibilities: 1 (, ), xi is rational number is countable subset. Convergence is to introduce metric spaces and give some examples in Section 1 each C 2 C. Choose a. metric! Suppose Sis open and closed are and ∅ in 3 dimensions if )! To consider a subset C of a set is equal to its closure on X it!
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closure of a set examples in metric space 2020