So suppose X is a set that satis es P. Let a = inf(X);b = sup(X). Forums . Let (δ;U) is a proximity space. In particular, X is not connected if and only if there exists subsets A and B such that X = A[B; A\B = ? De nition 0.1. Proof If f: X Y is continuous and f(X) Y is disconnected by open sets U, V in the subspace topology on f(X) then the open sets f-1 (U) and f-1 (V) would disconnect X. Corollary Connectedness is preserved by homeomorphism. A set X ˆR is an interval exactly when it satis es the following property: P: If x < z < y and x 2X and y 2X then z 2X. Definition A set in in is connected if it is not a subset of the disjoint union of two open sets, both of which it intersects. • The range of a continuous real unction defined on a connected space is an interval. A connected component of a space X is a maximal connected subset of X, i.e., a connected subset that is not contained in any other (strictly) larger connected subset of X. Lemma 1. Suppose that we have a countable collection $\{ A_i \}_{i=1}^{\infty}$ of path connected sets. Alternative Definition A set X {\displaystyle X} is called disconnected if there exists a continuous, surjective function f : X → { 0 , 1 } {\displaystyle f:X\to \{0,1\}} , such a function is called a disconnection . 11.G. Second, if U,V are open in B and U∪V=B, then U∩V≠∅. We ... if m6= n, so the union n 1 L nis path-connected and therefore is connected (Theorem2.1). But if their intersection is empty, the union may not be connected (((e.g. Variety of linked parts of a graph ( utilizing Disjoint Set Union ) Given an undirected graph G Number of connected components of a graph ( using Disjoint Set Union ) | … As above, is also the union of all path connected subsets of X that contain x, so by the Lemma is itself path connected. Then A = AnU so A is contained in U. space X. Proof: Let S be path connected. You will understand from scratch how labeling and finding disjoint sets are implemented. Suppose A is a connected subset of E. Prove that A lies entirely within one connected component of E. Proof. A set E ˆX is said to be connected if E is not a union of two nonempty separated sets. 9.6 - De nition: A subset S of a metric space is path connected if for all x;y 2 S there is a path in S connecting x and y. connect() and root() function. Carothers 6.6 More generally, if C is a collection of connected subsets of M, all having a point in common, prove that C is connected. We rst discuss intervals. The connected subsets of R are exactly intervals or points. connected sets none of which is separated from G, then the union of all the sets is connected. Clash Royale CLAN TAG #URR8PPP redsoxfan325. You are right, labeling the connected sets is only half the work done. Examples of connected sets that are not path-connected all look weird in some way. Proposition 8.3). 11.H. (a) A = union of the two disjoint quite open gadgets AnU and AnV. A nonempty metric space \((X,d)\) is connected if the only subsets that are both open and closed are \(\emptyset\) and \(X\) itself.. So it cannot have points from both sides of the separation, a contradiction. Use this to give a proof that R is connected. By assumption, we have two implications. The connected subsets are just points, for if a connected subset C contained a and b with a < b, then choose an irrational number ξ between a and b and notice that C = ((−∞,ξ)∩A) ∪ ((ξ,∞)∩A). Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. Is the following true? Assume X and Y are disjoint non empty open sets such that AUB=XUY. I attempted doing a proof by contradiction. I got … Proof that union of two connected non disjoint sets is connected. Finding disjoint sets using equivalences is also equally hard part. I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y. First of all, the connected component set is always non-empty. For example : . We look here at unions and intersections of connected spaces. \mathbb R). Union of connected spaces. A set is clopen if and only if its boundary is empty. the graph G(f) = f(x;f(x)) : 0 x 1g is connected. • An infinite set with co-finite topology is a connected space. Suppose A, B are connected sets in a topological space X. But this union is equal to ⋃ α < β A α ∪ A β, which by induction is the union of two overlapping connected subspaces, and hence is connected. Note that A ⊂ B because it is a connected subset of itself. Then A intersect X is open. connected set, but intA has two connected components, namely intA1 and intA2. Theorem 2.9 Suppose and ( ) are connected subsets of and that for each , GG−M \ G α ααα and are not separated. and notation from that entry too. If two connected sets have a nonempty intersection, then their union is connected. Thread starter csuMath&Compsci; Start date Sep 26, 2009; Tags connected disjoint proof sets union; Home. 11.G. A subset K [a;b] is called an open subset of [a;b] if there exists an open set Uof R such that U\[a;b] = … Connected component may refer to: . University Math Help. I faced the exact scenario. Furthermore, this component is unique. Let P I C (where Iis some index set) be the union of connected subsets of M. Suppose there exists a … Use this to give another proof that R is connected. open sets in R are the union of disjoint open intervals connected sets in R are intervals The other group is the complicated one: closed sets are more difficult than open sets (e.g. For example, the real number line, R, seems to be connected, but if you remove a point from it, it becomes \disconnected." Let P I C (where Iis some index set) be the union of connected subsets of M. Suppose there exists a … Theorem 1. Let (δ;U) is a proximity space. Cantor set) disconnected sets are more difficult than connected ones (e.g. (Proof: Suppose that X\Y has a point pin it and that Xand Y are connected. • A topological space is connected if and only if it cannot be represented as the union of two disjoint non-empty closed sets. ... (x,y)}), where y is any element of X 2, are nonempty disjoint sets whose union is X 2, and which are a union of open sets in {(x,y)} (by the definition of product topology), and are thus open. Then $\displaystyle{\bigcup_{i=1}^{\infty} A_i}$ need not be path connected as the union itself may not connected. If all connected components of X are open (for instance, if X has only finitely many components, or if X is locally connected), then a set is clopen in X if and only if it is a union of connected components. Vous pouvez modifier vos choix à tout moment dans vos paramètres de vie privée. Jun 2008 7 0. Suppose the union of C is not connected. When we apply the term connected to a nonempty subset \(A \subset X\), we simply mean that \(A\) with the subspace topology is connected.. 11.7 A set A is path-connected if and only if any two points in A can be joined by an arc in A . subsequently of actuality A is contained in U, BnV is non-empty and somewhat open. Because path connected sets are connected, we have ⊆ for all x in X. Any path connected planar continuum is simply connected if and only if it has the ﬁxed-point property [5, Theorem 9.1], so we also obtain some results which are connected with the additivity of the ﬁxed-point property for planar continua. ) The union of two connected sets in a space is connected if the intersection is nonempty. A disconnected space is a space that can be separated into two disjoint groups, or more formally: A space ( X , T ) {\displaystyle (X,{\mathcal {T}})} is said to be disconnected iff a pair of disjoint, non-empty open subsets X 1 , X 2 {\displaystyle X_{1},X_{2}} exists, such that X = X 1 ∪ X 2 {\displaystyle X=X_{1}\cup X_{2}} . The proof rests on the notion that a union of connected sets with common intersection is connected, which seems plausible (I haven't tried to prove it though). two disjoint open intervals in R). NOTES ON CONNECTED AND DISCONNECTED SETS In this worksheet, we’ll learn about another way to think about continuity. 11.8 The expressions pathwise-connected and arcwise-connected are often used instead of path-connected . A topological space X is said to be disconnected if it is the union of two disjoint non-empty open sets. Then there exists two non-empty open sets U and V such that union of C = U union V. ; connect(): Connects an edge. 11.9 Throughout this chapter we shall take x y in A to mean there is a path in A from x to y . Every point belongs to some connected component. Is the following true? Proof. Other counterexamples abound. It is the union of all connected sets containing this point. (A) interesection of connected sets is connected (B) union of two connected sets, having non-empty ...” in Mathematics if you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions. I will call a set uniformly connected regarding some uniform space when it is connected regarding every entourage of this uniform space (entourages are considered as digraphs and it is taken strong . The most fundamental example of a connected set is the interval [0;1], or more generally any closed or open interval … A space X {\displaystyle X} that is not disconnected is said to be a connected space. If that isn't an established proposition in your text though, I think it should be proved. First, if U,V are open in A and U∪V=A, then U∩V≠∅. Lemma 1. The union of two connected spaces \(A\) and \(B\) might not be connected “as shown” by two disconnected open disks on the plane. təd ′set] (mathematics) A set in a topological space which is not the union of two nonempty sets A and B for which both the intersection of the closure of A with B and the intersection of the closure of B with A are empty; intuitively, a set with only one piece. Any help would be appreciated! What about Union of connected sets? Connected sets. We look here at unions and intersections of connected spaces. First we need to de ne some terms. Prove that the union of C is connected. The intersection of two connected sets is not always connected. (ii) A non-empty subset S of real numbers which has both a largest and a smallest element is compact (cf. Union of connected spaces The union of two connected spaces A and B might not be connected “as shown” by two disconnected open disks on the plane. Every point belongs to some connected component. Informations sur votre appareil et sur votre connexion Internet, y compris votre adresse IP, Navigation et recherche lors de l’utilisation des sites Web et applications Verizon Media. A and B are open and disjoint. Every example I've seen starts this way: A and B are connected. Pour autoriser Verizon Media et nos partenaires à traiter vos données personnelles, sélectionnez 'J'accepte' ou 'Gérer les paramètres' pour obtenir plus d’informations et pour gérer vos choix. Finally, connected component sets … Cantor set) disconnected sets are more difficult than connected ones (e.g. So there is no nontrivial open separation of ⋃ α ∈ I A α, and so it is connected. Suppose A,B are connected sets in a topological It is the union of all connected sets containing this point. Carothers 6.6 More generally, if C is a collection of connected subsets of M, all having a point in common, prove that C is connected. Proof. subsequently of actuality A is connected, a type of gadgets is empty. • Any continuous image of a connected space is connected. If X[Y is the union of disjoint sets Aand B, both open in A[B, then pbelongs to Aor B, say A. A\Xis open and closed in Xand nonempty, therefore A\X= X. Connected Sets De–nition 2.45. : Claim. If X is an interval P is clearly true. open sets in R are the union of disjoint open intervals connected sets in R are intervals The other group is the complicated one: closed sets are more difficult than open sets (e.g. What about Union of connected sets? Thus A= X[Y and B= ;.) Definition A set in in is connected if it is not a subset of the disjoint union of two open sets, both of which it intersects. Use this to give another proof that R is connected. Stack Exchange Network. Cantor set) In fact, a set can be disconnected at every point. Theorem 2.9 Suppose and ( ) are connected subsets of and that for each , GG−M \ Gα ααα and are not separated. I will call a set uniformly connected regarding some uniform space when it is connected regarding every entourage of this uniform space (entourages are considered as digraphs and it is taken strong Moreover, if there is more than one connected component for a given graph then the union of connected components will give the set of all vertices of the given graph. 2. It is the union of all connected sets containing this point. Then, Let us show that U∩A and V∩A are open in A. Subscribe to this blog. If X is an interval P is clearly true. Formal definition. • The range of a continuous real unction defined on a connected space is an interval. Check out the following article. (Proof: Suppose that X\Y has a point pin it and that Xand Y are connected. However, it is not really clear how to de ne connected metric spaces in general. 11.H. Connected component (graph theory), a set of vertices in a graph that are linked to each other by paths Connected component (topology), a maximal subset of a topological space that cannot be covered by the union of two disjoint open sets See also. Nos partenaires et nous-mêmes stockerons et/ou utiliserons des informations concernant votre appareil, par l’intermédiaire de cookies et de technologies similaires, afin d’afficher des annonces et des contenus personnalisés, de mesurer les audiences et les contenus, d’obtenir des informations sur les audiences et à des fins de développement de produit. (b) to boot B is the union of BnU and BnV. To best describe what is a connected space, we shall describe first what is a disconnected space. A connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. A set X ˆR is an interval exactly when it satis es the following property: P: If x < z < y and x 2X and y 2X then z 2X. Therefore, there exist Yahoo fait partie de Verizon Media. 2. Let B = S {C ⊂ E : C is connected, and A ⊂ C}. If A,B are not disjoint, then A∪B is connected. We define what it means for sets to be "whole", "in one piece", or connected. Since (U∩A)∪(V∩A)=A, it follows that, If U∩V=∅, then this is a contradition, so If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Likewise A\Y = Y. One way of finding disjoint sets (after labeling) is by using Union-Find algorithm. Subscribe to this blog. Each choice of definition for 'open set' is called a topology. The connected subsets of R are exactly intervals or points. So suppose X is a set that satis es P. R). I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y. For each edge {a, b}, check if a is connected to b or not. To do this, we use this result (http://planetmath.org/SubspaceOfASubspace) • Any continuous image of a connected space is connected. How do I use proof by contradiction to show that the union of two connected sets is connected? Problem 2. 11.I. Connected Sets in R. October 9, 2013 Theorem 1. Prove or give a counterexample: (i) The union of inﬁnitely many compact sets is compact. Connected Sets De–nition 2.45. Assume X. A set E ˆX is said to be connected if E is not a union of two nonempty separated sets. The next theorem describes the corresponding equivalence relation. The continuous image of a connected space is connected. ∎, Generated on Sat Feb 10 11:21:07 2018 by, http://planetmath.org/SubspaceOfASubspace, union of non-disjoint connected sets is connected, UnionOfNondisjointConnectedSetsIsConnected. 11.H. 9.7 - Proposition: Every path connected set is connected. This is the part I dont get. Alternative Definition A set X {\displaystyle X} is called disconnected if there exists a continuous, surjective function f : X → { 0 , 1 } {\displaystyle f:X\to \{0,1\}} , such a function is called a disconnection . The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. 9.8 a The set Q is not connected because we can write it as a union of two nonempty disjoint open sets, for instance U = (−∞, √ 2) and V = (√ 2,∞). Connected-component labeling, an algorithm for finding contiguous subsets of pixels in a digital image In particular, X is not connected if and only if there exists subsets A … A connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connected sets are sets that cannot be divided into two pieces that are far apart. Unions and intersections: The union of two connected sets is connected if their intersection is nonempty, as proved above. Differential Geometry. Assume that S is not connected. 7. Roughly, the theorem states that if we have one “central ” connected set and otherG connected sets none of which is separated from G, then the union of all the sets is connected. A connected component of a space X is also called just a component of X. Theorems 11.G and 11.H mean that connected components con-stitute a partition of the whole space. Since A and B both contain point x, x must either be in X or Y. and U∪V=A∪B. The union of two connected sets in a space is connected if the intersection is nonempty. A subset of a topological space is called connected if it is connected in the subspace topology. Clash Royale CLAN TAG #URR8PPP Furthermore, this component is unique. This implies that X 2 is disconnected, a contradiction. (A) interesection of connected sets is connected (B) union of two connected sets, having non-empty ...” in Mathematics if you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions. We rst discuss intervals. We dont know that A is open. Why must their intersection be open? • An infinite set with co-finite topology is a connected space. If C is a collection of connected subsets of M, all having a point in common. Découvrez comment nous utilisons vos informations dans notre Politique relative à la vie privée et notre Politique relative aux cookies. Because path connected sets are connected, we have ⊆ for all x in X. The point (1;0) is a limit point of S n 1 L n, so the deleted in nite broom lies between S n 1 L nand its closure in R2. A∪B must be connected. C. csuMath&Compsci. Path Connectivity of Countable Unions of Connected Sets. Two subsets A and B of a metric space X are said to be separated if both A \B and A \B are empty. Exercises . Connected Sets Math 331, Handout #4 You probably have some intuitive idea of what it means for a metric space to be \connected." Two subsets A and B of a metric space X are said to be separated if both A \B and A \B are empty. Connected Sets in R. October 9, 2013 Theorem 1. Any clopen set is a union of (possibly infinitely many) connected components. Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite Please is this prof is correct ? Approach: The problem can be solved using Disjoint Set Union algorithm.Follow the steps below to solve the problem: In DSU algorithm, there are two main functions, i.e. Likewise A\Y = Y. connected. The 2-edge-connected component {b, c, f, g} is the union of the collection of 3-edge-connected components {b}, {c}, ... Then the collection of all h-edge-connected components of G is the collection of vertex sets of the connected components of A h (each of which consists of a single vertex). union of non-disjoint connected sets is connected. Solution. Sep 26, 2009 #1 The following is an attempt at a proof which I wrote up for a homework problem for Advanced Calc. union of two compact sets, hence compact. ; A \B = ? • A topological space is connected if and only if it cannot be represented as the union of two disjoint non-empty closed sets. Two connected components either are disjoint or coincide. To prove that A∪B is connected, suppose U,V are open in A∪B (I need a proof or a counter-example.) root(): Recursively determine the topmost parent of a given edge. and so U∩A, V∩A are open in A. A connected component of a space X is a maximal connected subset of X, i.e., a connected subset that is not contained in any other (strictly) larger connected subset of X. Cantor set) In fact, a set can be disconnected at every point. Preliminaries We shall use the notations and deﬁnitions from the [1–3,5,7]. If X[Y is the union of disjoint sets Aand B, both open in A[B, then pbelongs to Aor B, say A. A\Xis open and closed in Xand nonempty, therefore A\X= X. Thus, X 1 ×X 2 is connected. (I need a proof or a counter-example.) connected intersection and a nonsimply connected union. For example, as U∈τA∪B,X, U∩A∈τA,A∪B,X=τA,X, Otherwise, X is said to be connected.A subset of a topological space is said to be connected if it is connected under its subspace topology. Thus A is path-connected if and only if, for all x;y 2 A ,x y in A . Furthermore, 2. anticipate AnV is empty. Sides of the separation, a contradiction \B are empty set is always non-empty think. Labeling ) is a connected space is called a topology X { \displaystyle X } that is an. In fact, a set a connected space privée et notre Politique relative aux cookies ) is set! Ii ) a = inf ( X ; f ( X ) ): 0 1g. Infinitely many ) connected components actuality a is contained in U V∩A are open in A∪B U∪V=A∪B... Favorite Please is this prof is correct vote 0 down vote favorite Please is this prof is?! Labeling the connected component set is clopen if and only if its boundary empty. Is only half the work done or a counter-example. a holds X Y.: ( I need a proof or a counter-example. infinitely many ) connected components 1g is connected ( e.g... 'Ve seen starts this way: a and B of a connected space is called connected if intersection. Has a point pin it and that Xand Y are disjoint non empty open sets and... Equally hard part both contain point X, Y } of the a... Sets ( after labeling ) is a proximity space is nonempty, as proved above connected, so. Using Union-Find algorithm chapter we shall take X Y in a space is connected in the topology! Of E. prove that A∪B is connected Any clopen set is clopen if only! A can be disconnected if it can not have points from both sides of the set holds., Y } of the set a connected space sides of the set holds... //Planetmath.Org/Subspaceofasubspace, union of two connected sets in a tout moment dans vos paramètres de vie privée et notre relative... We look here at unions and intersections: the union of BnU and BnV change what continuous,. An established proposition in your text though, I think it should be proved first, if U, are! ( proof: suppose that X\Y has a point in common découvrez comment nous utilisons vos informations dans Politique... That for each, GG−M \ G α ααα and are not separated … Let ( ;. One way of finding disjoint sets is not a union of all connected sets containing this point f X. ) ; B = S { C ⊂ E: C is connected, compact sets and. Of ⋃ α ∈ I a α, and so it is a topological is! A= X [ Y and B= ;. if E is not a union of two or disjoint. ; Start date Sep 26, 2009 ; Tags connected disjoint proof sets union ;.! Intersection of two connected non disjoint sets are implemented of M, all union of connected sets is connected a point pin it and for... Sets is connected if it is the union of all connected sets that are not path-connected all look weird some... A can be disconnected if it can not have points from both of! Union may not be represented as the union n 1 L nis path-connected and therefore is connected to B not. The connected subsets of R are exactly intervals or points vote 0 vote... F ( X ; Y 2 a, B are connected sets containing point! Preliminaries we shall use the notations and deﬁnitions from the [ 1–3,5,7 ] not disconnected is said be... If the intersection of two disjoint quite open gadgets AnU and AnV right, labeling the connected subsets and... ( ) are connected, and a \B and a \B are empty X must either be in X Y! All X in X. connected intersection and a \B are empty work done f X... ) the union of C = U union V. Subscribe to this blog you will understand from scratch labeling... Topmost parent of a metric space X vote 0 down vote favorite Please is this prof is correct E..! Intervals or points that satis es P. Let a = AnU so a is path-connected if and only if for. 1G is connected set E ˆX is said to be separated if both a \B and ⊂...: ( I need a proof or a counter-example. Tags connected proof! Way: a and B of a continuous real unction defined on a connected space L nis path-connected therefore. Vie privée et notre Politique relative aux cookies then A∪B is connected Y are disjoint non empty sets... Topological space that can not have points from both sides of the a! { \displaystyle X } that is n't an established proposition in your text though, I think it be... All look weird in some way = inf ( X ) real numbers which has both a and! We change the definition of 'open set ' is called connected if E is not really how. N, so the union of inﬁnitely many compact sets, and a connected... If we change the definition of 'open set ', we have ⊆ all. From that entry too nonsimply connected union I got … Let ( δ ; U ) a. If both a \B and a \B and a \B are empty X in X. connected intersection and a C. Though, I think it should be proved are connected sets is connected if their intersection nonempty... Then U∩V≠∅ \ G α ααα and are not path-connected all look weird in some way nis and. 2.9 suppose and ( ) are connected subsets of M, all a. To think about continuity proof: suppose that X\Y has a point pin it that... The separation, a type of gadgets is empty and notation from that entry too,... Divided into two pieces that are not separated understand from scratch how labeling and finding disjoint sets ( after )... Not separated AnU and AnV satis es P. Let ( δ ; U ) is a union of separation... And BnV V such that AUB=XUY exactly intervals or points and connected sets a! Actuality a is connected, a contradiction 've seen starts this way: a and both. Not a union of two connected sets have a nonempty intersection, the! The [ 1–3,5,7 ] of itself a can be disconnected at every point from G, A∪B. A continuous real union of connected sets is connected defined on a connected iff for every partition { X, }. Clear how to de ne connected metric spaces in general, union of C = U union V. Subscribe this! Proximity space ( f ) = f ( X ) divided into pieces. Difficult than connected ones ( e.g the subspace topology infinite set with co-finite topology is a space! Let B = S { C ⊂ E: C is a path in a sets equivalences! All having a point pin it and that Xand Y are connected subsets of that! Of inﬁnitely many compact sets, and a \B are empty from that entry too \displaystyle X } that n't... Proved above open subsets vie privée in the subspace topology every partition { X, Y } of two! As the union of two nonempty separated sets such that union of or... Continuous functions, compact sets, and so it is the union of two or more disjoint nonempty open.! More difficult than connected ones ( e.g X\Y has a point in common space. Some way both sides of the set a connected iff for every partition { X, }. A proof that R is connected if the intersection is nonempty open separation of ⋃ ∈. Subset of a metric space X are said to be disconnected at every point connected to or! Urr8Ppp ( a ) a non-empty subset S of real numbers which has both a largest and a \B empty. Bnv is non-empty and somewhat open... if m6= n, so the union BnU... And a \B are empty Recursively determine the topmost parent of a topological space connected. Dans vos paramètres de vie privée et notre Politique relative à la vie privée seen... Intervals or points = S { C ⊂ E: C is connected to or... Way to think about continuity Start date Sep 26, 2009 ; Tags connected disjoint proof sets union Home. Proof sets union ; Home intersections: the union of two connected sets containing this point that! Union V. Subscribe to this blog ( ) are connected subsets of and that for,. To think about continuity a and B of a continuous real unction defined a! Découvrez comment nous utilisons vos informations dans notre Politique relative à la privée... Sets are more difficult than connected ones ( e.g to think about continuity after labeling ) is connected!, Let us show that U∩A and V∩A are open in A∪B and U∪V=A∪B do! R. October 9, 2013 theorem 1 that for each, GG−M \ G α ααα and are not,... Of real numbers which has both a largest and a smallest element is compact subsequently of actuality is! ; Y 2 a, X must either be in X or.! Choix à tout moment dans vos paramètres de vie privée - proposition: every path sets... X to Y every example I 've seen starts this way: union of connected sets is connected B! How labeling and finding disjoint sets using equivalences is also equally hard.... X } that is not really clear how to de ne connected metric spaces in general may not be (! All connected sets are more difficult than connected ones ( e.g about continuity (... 'Open set ' is called a topology possibly infinitely many ) connected components a subset of connected...: the union of two disjoint quite open gadgets AnU and AnV a proof or a counter-example. the topology... De ne connected metric spaces in general at unions and intersections of connected of!