/pgf@ca0.25 << /F61 40 0 R For example, when these boundaries are blurred, the children often become the parent to the parents. 20 0 obj /CA 0.7 >> �� ��C]��R���``��1^,"L),���>�xih�@I9G��ʾ�8�1�Q54r�mz�o��Ȑ����l5_�1����^����m ͑�,�W�T�h�.��Z��U�~�i7+��n-�:���}=4=vx9$��=��5�b�I�������63�a�Ųh�\�y��3�V>ڥ��H����ve%6��~�E�prA����VD��_���B��0F9��MW�.����Q1�&���b��:;=TNH��#)o _ۈ}J)^?N�N��u��Ez��v|�UQz���AڡD�o���jaw.�:E�VB ���2��|����2[D2�� << Precision perimeter Eclosure 0.182 ft. 939.46 ft. 1 5,176 Side Length (ft.) Latitude Departure degree minutes AB S 6 15 W 189.53 -188.403 -20.634 BC S 29 38 E 175.18 -152.268 86.617 CD N 81 18 W 197.78 29.916 -195.504 https://goo.gl/JQ8Nys Finding the Interior, Exterior, and Boundary of a Set Topology De nition 1.1. /ItalicAngle 0 Posted on December 2, 2020 by December 2, 2020 by /F33 28 0 R Videos for the course MTH 427/527 Introduction to General Topology at the University at Buffalo. >> 3 0 obj << /pgf@ca1 << /ca 0.2 Let S be an arbitrary set in the real line R. A point b R is called boundary point of S if every non-empty neighborhood of b intersects S and the complement of S. The set of all boundary points of S is called the boundary of S, denoted by bd(S). {\displaystyle \mathbb {R} ^ {2}} , the boundary of a closed disk. endobj endobj The set of boundary points is called the boundary of A and is denoted by ! /ca 0.8 Within each type, we can have three boundary states: 1.) Interior, exterior, limit, boundary, isolated point. /PaintType 2 >> /Annots [ 77 0 R 78 0 R ] Suppose T ˆE satis es S ˆT ˆS. Obviously, itsexterior is x2+ y2+z2> 1. Example 3.2. endstream /Type /Page Interior and Boundary Points of a Set in a Metric Space. Show transcribed image text. /YStep 2.98883 A. A= N(-2+1,2+ =) NEN IntA= Bd A= CA= A Is Closed / Open / Neither Closed Nor Open B. You should change all open balls to open disks. << 3 0 obj b) Given that U is the set of interior points of S, evaluate U closure. /MediaBox [ 0 0 612 792 ] >> Consider a sphere, x2+ y2+ z2= 1. 17 0 obj That is the closure design principle in action! De nition { Neighbourhood Suppose (X;T) is a topological space and let x2Xbe an arbitrary point. For a general metric space, the closed ball ˜Br(x0): = {x ∈ X: d(x, x0) ≤ r} may be larger than the closure of a ball, ¯ Br(x0). - the boundary of Examples. ies: a theoretical line that marks the limit of an area of land Merriam Webster’s Dictionary of Law. >> /pgf@ca0.5 << Latitudes and Departures - Example 22 EEEclosure L D 0.079 0.16322 0.182 ft. /Parent 1 0 R >> /Resources 58 0 R iff iff /XStep 2.98883 endobj 11 0 obj a is an interior point of M, because there is an ε-neighbourhood of a which is a subset of M. In any space, the interior of the empty set is the empty set. /Contents 59 0 R 15 0 obj /FontFile 20 0 R Coverings. A definition of what boundaries ARE, examples of different types of boundaries, and how to recognize and define your own boundaries. endobj >> /pgf@CA.4 << Def. /Contents 57 0 R endobj %PDF-1.3 Theorem: A set A ⊂ X is closed in X iﬀ A contains all of its boundary points. Interiors, Closures, and Boundaries Brent Nelson Let (E;d) be a metric space, which we will reference throughout. This problem has been solved! Proof. The Boundary of a Set. The closure of D is. endobj << Closed sets have complementary properties to those of open sets stated in Proposition 5.4. 23) and compact (Sec. >> /pgf@ca0.3 << Point set. Interior, exterior and boundary points. By using our services, you agree to our use of cookies. >> These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure. /Resources 65 0 R >> 02. /Parent 1 0 R /Type /Pages endobj >> /Contents 64 0 R /Type /Page /Annots [ 81 0 R ] In any space X, if S ⊆ X, then int S ⊆ S. If X is the Euclidean space ℝ of real numbers, then int ( [0, 1]) = (0, 1). Interior of a set. /Length2 19976 /Kids [ 3 0 R 4 0 R 5 0 R 6 0 R 7 0 R 8 0 R 9 0 R 10 0 R ] Where training is possible, external boundaries can be replaced by internal ones. zPressure inlet boundary is treated as loss-free transition from stagnation to inlet conditions. The Boundary of a Set in a Topological Space; The Boundary of a Set in a Topological Space Examples 1; The Boundary of Any Set is Closed in a Topological Space << Set N of all natural numbers: No interior point. Our current model is internal and the fluid is bound by the pipe walls. /MediaBox [ 0 0 612 792 ] Ł�*�l��t+@�%\�tɛ]��ӏN����p��!���%�W��_}��OV�y�k� ���*n�kkQ�h�,��7��F.�8
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�Ŏkϝ�6Ou��=��j����.er�Й0����7�UP�� p� A and ! • The complement of A is the set C(A) := R \ A. 18), connected (Sec. zFLUENT calculates static pressure and velocity at inlet zMass flux through boundary varies depending on interior solution and specified flow direction. Math 3210-3 HW 10 Solutions NOTE: You are only required to turn in problems 1-5, and 8-9. Rigid boundaries, which are too strong, can be likened to walls without doors. Unlike the convex hull, the boundary can shrink towards the interior of the hull to envelop the points. Interior, Closure, Boundary 5.1 Deﬁnition. Exercise: Show that a set S is an open set if and only if every point of S is an interior point. Its interioris the set of all points that satisfyx2+ y2+ z2 1, while its closure is x2+ y2+ z2= 1. In these exercises, we formalize for a subset S ˆE the notion of its \interior", \closure", and \boundary," and explore the relations between them. 1 Interior, closure, and boundary Recall the de nitions of interior and closure from Homework #7. Examples 5.1.2: Which of the following sets are open, closed, both, or neither ? The other “universally important” concepts are continuous (Sec. The example above shows 4 squares and over them is a white circle. endobj /pgf@ca.7 << Bounded, compact sets. 6 0 obj /MediaBox [ 0 0 612 792 ] /pgf@ca0.6 << A relic boundary is one that no longer functions but can still be detected on the cultural landscape. /MediaBox [ 0 0 612 792 ] �������`�9�L-M\��5�����vf�D�����ߔ�����T�T��oL��l~��`��],M T�?���` Wy#[ ���?��l-m~����5 ��.T��N�F6��Y:KXz L-]L,�K��¥]�l,M���m ��fg Definition 5.1.5: Boundary, Accumulation, Interior, and Isolated Points : Let S be an arbitrary set in the real line R.. A point b R is called boundary point of S if every non-empty neighborhood of b intersects S and the complement of S.The set of all boundary points of S is called the boundary of S, denoted by bd(S). of A nor an interior point of X \ A . Definition. (By the way, a closed set need not have any boundary points at all: in $\Bbb R$ the only examples of this phenomenon are the closed sets $\varnothing$ and $\Bbb R$, but in more general topological spaces there can be many sets that are simultaneously open and closed and which therefore have empty boundary.) << Or, equivalently, the closure of solid S contains all points that are not in the exterior of S. Examples Here is an example in the plane. /BBox [ -0.99628 -0.99628 3.9851 3.9851 ] I= (0;1] isn’t closed since, for example, (1=n) is a convergent sequence in Iwhose limit 0 doesn’t belong to I. (c)For E = R with the usual metric, give examples of subsets A;B ˆR such that A\B 6= A \B and (A[B) 6= A [B . 01. Table of Contents. << Boundary of a set De nition { Boundary Suppose (X;T) is a topological space and let AˆX. Interior, Closure, Exterior and Boundary Let (X;d) be a metric space and A ˆX. |||||{Solutions: x�+T0�3��0U(2��,-,,�r��,,L�t��fF /F39 46 0 R For all of the sets below, determine (without proof) the interior, boundary, and closure of each set. /pgf@ca.4 << endobj (In t A ) " ! /CA 0.4 >> /Resources 67 0 R /ca 0.6 /Ascent 696 >> /Type /Page endobj /CA 0.25 /PatternType 1 18), homeomorphism (Sec. Unreviewed /FontBBox [ -350 -309 1543 1127 ] /pgf@CA0.2 << >> R 2. /TilingType 1 f1g f1g [0;1) (0;1) [0;1] f0;1g (0;1)[(1;2) (0;1)[(1;2) [0;2] f0;1;2g [0;1][f2g (0;1) [0;1][f2g f0;1;2g Z ? For each of the following subsets of R2, decide whether it is open, closed, both or neither. The points may be points in one, two, three or n-dimensional space. Classify It As Open, Closed, Or Neither Open Nor Closed. Examples of … Open, Closed, Interior, Exterior, Boundary, Connected For maa4402 January 1, 2017 These are a collection of de nitions from point set topology. /pgf@ca0.4 << Bounded, compact sets. The boundary of Ais de ned as the set @A= A\X A. << >> >> << << << Show that T is also connected. /Contents 62 0 R /Encoding 22 0 R �+ � S = fz 2C : jzj= 1g, the unit circle. /Length 53 De–nition Theinteriorof A, denoted intA, is the largest open set contained in A (alternatively, the union of all open sets contained in A). If both Aand its complement is in nite, then arguing as above we see that it has empty interior and its closure is X. 03. Let (X;T) be a topological space, and let A X. 5 0 obj << �5ߊi�R�k���(C��� >> 12 0 obj � >> or U= RrS where S⊂R is a ﬁnite set. Z Z Q ? /CA 0.6 /Resources << Derived set. We give some examples based on the sets collected below. stream - the exterior of . Expert Answer 100% (1 rating) Previous question Next question Transcribed Image Text from this Question. /CA 0.5 )#��I�St`�bj�JBXG���֖���9������)����[�H!�Jt;�iR�r"��9&�X�-�58XePԫ��c!���[��)_b�0���@���_%M�4dˤ��Hۛ�H�G�m
���3�槔`��>8@�]v�6�^!�����n��o�,J We made a [boundary] of trees at the back of our… ��˻|�ctK��S2,%�F. /FontDescriptor 19 0 R Def. Basic Theorems Regarding the Closure of Sets in a Topological Space; A Comparison of the Interior and Closure of a Set in a Topological Space; 2.5. (a) /CA 0 /Filter /FlateDecode /F48 53 0 R /Descent -206 Examples of … >> << /Filter /FlateDecode We then add the fluid we are simulating to the project. /F45 37 0 R Content: 00:00 Page 46: Interior, closure, boundary: definition, and first examples… /Resources 13 0 R Consider the subset A= Q R. For example, given the usual topology on. ��I��%��Q�i���W��s�R� ՝%��^�����*Z�7�R��s��։E%fE%�Clp,+�Y ������r�}�� Z���p�:l�Iߗt�m+n�T���rS��^��)DIw�����! Proof. If anyone could explain interior and closure sets like I'm a five year old, and be prepared for dumb follow-up questions, I would really appreciate it. (Interior of a set in a topological space). Figure 5. /ca 0.7 /Parent 1 0 R Find Interior, Boundary And Closure Of A-{x ; Question: Find Interior, Boundary And Closure Of A-{x . Interior and Boundary Points ofa Region in the Plane x1 x2 0 c a B 1.4. /Parent 1 0 R >> Each row of k is a triangle defined in terms of the point indices. Interior point. >> /F59 23 0 R /pgf@ca0.7 << Cookies help us deliver our services. Exterior points: If a point is not an interior point or boundary point of S, it is an exterior point of S. Lecture 2 Open and Closed set. For any set S, ∂S⊇∂∂S, with equality holding if and only if the boundary of Shas no interior points, which will be the case for example if Sis either closed or open. >> If we let X be a space with the discrete metric, {d(x, x) = 0, d(x, y) = 1, x ≠ y. Interior, exterior and boundary points. /Count 8 Interior points of regions in space (R3). /Length1 980 Point set. Some of these examples, or similar ones, will be discussed in detail in the lectures. /ca 1 1996. boundary I endobj Math 104 Interiors, Closures, and Boundaries Solutions (b)Show that (A\B) = A \B . >> /ExtGState 17 0 R From Wikibooks, open books for an open world < Real AnalysisReal Analysis. /Parent 1 0 R Let Xbe a topological space. Selecting the analysis type. Theorems. Remark: The interior, exterior, and boundary of a set comprise a partition of the set. De–nition Theclosureof A, denoted A , is the smallest closed set containing A /Annots [ 61 0 R ] R R R R R ? /LastChar 124 Defining the project fluids. /pgf@CA0.6 << >> The closure of A is the union of the interior and boundary of A, i.e. A set whose elements are points. Theorems. See the answer. In the second video, we will explore how to set boundaries, which includes communicating your boundaries to others. Closure of a set. If A= [ 1;1] ( 1;1) inside of X= R2, then @A= A int(A) consists of points (x;y) on the edge of the unit square: it is equal to (f 1;1g [ 1;1]) [ ([ 1;1] f 1;1g); as you should check (from our earlier determination of the closure and interior of A). Ask Question Asked 6 years, 7 months ago. >> Interior and Boundary Points of a Set in a Metric Space Fold Unfold. /CA 0.4 endobj Let T Zabe the Zariski topology on R. Recall that U∈T Zaif either U= ? a nite complement, it is open, so its interior is itself, but the only closed set containing it is X, so its boundary is equal to XnA. 13 0 obj A closed interval [a;b] ⊆R is a closed set since the set Rr[a;b] = (−∞;a)∪(b;+∞) is open in R. 5.3 Example. stream An entire metric space is both open and closed (its boundary is empty). >> Can be used as a “free” boundary in an external or unconfined flow. �06l��}g �i���X%ﭟ0| YC��m�. /pgf@ca0.2 << Therefore, the closure is theunion of the interior and the boundary (its surfacex2+ y2+z2= 1). /FirstChar 27 is open iff is closed. ; A point s S is called interior point of S if there exists a neighborhood of s completely contained in S. One example is the Berlin Wall, which was built in 1961 by Soviet controlled East Germany to contain the portion of the city that had been given over to America, England, and France to administer. /pgfprgb [ /Pattern /DeviceRGB ] One warning must be given. >> Math 396. In l∞, B1 ∌ (1 / 2, 2 / 3, 3 / 4, …) ∈ ¯ B1. stream Perfect set. ����t���9������^m��-/,��USg�o,�� 2 0 obj /Pages 1 0 R [1] Franz, Wolfgang. There is no border existing as a separating line. The interior and exterior are both open, and the boundary is closed. /Filter /FlateDecode 1 De nitions We state for reference the following de nitions: De nition 1.1. Notice how the center of all 4 sides doesn’t touch, but your eye still completes the circle for you. /Parent 1 0 R For all of the sets below, determine (without proof) the interior, boundary, and closure of each set. 8. endobj k = boundary(x,y,z) returns a triangulation representing a single conforming 3-D boundary around the points (x,y,z). /pgf@ca0.8 << /ca 0.4 /Length 2303 Def. /ca 0.7 I first noticed it with dogs. endobj If has discrete metric, 2. >> Since the boundary of a set is closed, ∂∂S=∂∂∂S{\displaystyle \partial \partial S=\partial \partial \partial S}for any set S. /Parent 1 0 R A set whose elements are points. If is the real line with usual metric, , then Remarks. /Type /Page For example, imagine an area represented by a vector data model: it is composed of a border, which separates the interior from the exterior of the surface. << A . Interior and boundary points in space or R3. /ca 0.25 >> Anyone found skiing outside the [boundary] is putting himself in danger, and if caught, will lose his lift pass. /CA 0.3 An arbitrary intersection of closed sets is closed, and a nite union of closed sets is closed. I could continue to stare at definitions, but some human interaction would be a lot more helpful. /Type /Pattern >> >> /pgf@ca.3 << /XHeight 510 An external flow example would be airflow over an airplane wing. These are boundaries that define our family and make it distinctive from other families. For example, if X is the set of rational numbers, with the usual relative topology induced by the Euclidean space R, and if S = {q in Q : q 2 > 2, q > 0}, 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 … Interior and Boundary Points of a Set in a Metric Space. >> They are often impenetrable. %PDF-1.5 Uncategorized boundary math example. /pgf@CA0.5 << >> Distinguishing between fundamentally different spaces lies at the heart of the subject of topology, and it will occupy much of our time. 2. >> Solutions to Examples 3 1. Topology of the Reals 1. /Length3 0 >> /Contents 75 0 R Set Q of all rationals: No interior points. /Widths 21 0 R << example. /ColorSpace 14 0 R >> 8 0 obj `gJ�����d���ki(��G���$ngbo��Z*.kh�d�����,�O���{����e��8�[4,M],����������_����;���$��������geg"�ge�&bfgc%bff���_�&�NN;�_=������,�J x L`V�؛�[�������U��s3\Tah�$��f�u�b��� ���3)��e�x�|S�J4Ƀ�m��ړ�gL����|�|qą's��3�V�+zH�Oer�J�2;:��&�D��z_cXf���RIt+:6��3��9٠x� �t��u�|���E
��,�bL�@8��"驣��>�/�/!��n���e�H�����"�4z�dՌ�9�4. endobj /CA 0.8 /Type /Page /pgf@CA0.4 << << Example of a set whose boundary is not equal to the boundary of its closure. /pgfpat4 16 0 R << bdy G= cl G\cl Gc. /CapHeight 696 << Find The Boundary, The Interior, And The Closure Of Each Set. 10 0 obj 1. Def. Then intA = (0;1) [(2;3) A = [0;1] [[2;3] extA = int(X nA) = int ((1 ;0) [(1;2] [[3;+1)) = (1 ;0) [(1;2) [(3;+1) @A = (X nA) \A = ((1 ;0] [[1;2] [[3;+1)) \([0;1] [[2;3]) = f0;1;2;3g /ca 0.3 Example: The set {1,2,3,4,5} has no boundary points when viewed as a subset of the integers; on the other hand, when viewed as a subset of R, every element of the set is a boundary point. /Resources 76 0 R >> b) Given that U is the set of interior points of S, evaluate U closure. �wǮ�����p�x=��%�=�v�މ��K�A+�9��l�
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�ЕM)n�o|O���&��/����⻉�u~9�\wW�|s�/���7�&��]���;�}m~(���AF�1DcU�O|���3!N��#XSO�4��1�0J Some examples. p������>#�gff�N�������L���/ Some of these examples, or similar ones, will be discussed in detail in the lectures. Limit Points; Closure; Boundary; Interior; We are nearly ready to begin making some distinctions between different topological spaces. A topology on a set X is a collection τ of subsets of X, satisfying the following axioms: (1) The empty set and X are in τ (2) The union of any collection of sets in τ is also in τ (3) The intersection of any finite number of sets in τ is also in τ. Example 3.3. /Type /Page endobj /Contents 66 0 R /Length 1969 Derived Set, Closure, Interior, and Boundary We have the following deﬁnitions: • Let A be a set of real numbers. Table of Contents. Boundary of a boundary. Let S be an arbitrary set in the real line R.. A point b R is called boundary point of S if every non-empty neighborhood of b intersects S and the complement of S.The set of all boundary points of S is called the boundary of S, denoted by bd(S). /ca 0.4 boundary translation in English-Chinese dictionary. Interior and Boundary Points of a Set in a Metric Space. Limit points De nition { Limit point Let (X;T) be a topological space and let AˆX. Interior, closure and boundary: examples Theorem 2.6 { Interior, closure and boundary One has A \@A= ? /ca 0.5 Active 6 years, 7 months ago. endobj Regions. %���� << /F54 42 0 R /F42 32 0 R A point in the interior of A is called an interior point of A. "���J��m>�ZE7�������@���|��-�M�䇗{���lhmx:�d���
�ϻX����:��T�{�~��ý z��N >> In the space of rational numbers with the usual topology (the subspace topology of R), the boundary of (-\infty, a), where a is irrational, is empty. We use d(A) to denote the derived set of A, that is theset of all accumulation points of A.This set is sometimes denoted by A′. Note the diﬀerence between a boundary point and an accumulation point. /Producer (PyPDF2) Interior and Boundary Points of a Set in a Metric Space. Open and Closed Sets Definition 5.1.5: Boundary, Accumulation, Interior, and Isolated Points . >> Closure of a set. >> /Resources 80 0 R << 18 0 obj 11.Let S ˆE be a connected set. /Filter /FlateDecode Perfect set. D = fz 2C : jzj 1g, the closed unit disc. Please Subscribe here, thank you!!! Consider R2 with the Euclidean metric. /Contents 12 0 R 1.4.1. Family boundaries. 3.) /ca 0.3 Pro ve that for an y set A in a topological space we ha ve ! 9/20 . Limit Points, Closure, Boundary and Interior. This topology course is frying my brain. /Subtype /Type1 Thus, the algorithms implemented for vector data models are not valid for raster data models. Interior, closure, and boundary We wish to develop some basic geometric concepts in metric spaces which make precise certain intuitive ideas centered on the themes of “interior” and “boundary” of a subset of a metric space. The closure of a set also depends upon in which space we are taking the closure. >> 19 0 obj The closure of a solid S is defined to be the union of S's interior and boundary, written as closure(S). Ω = { ( x , y ) | x 2 + y 2 ≤ 1 } {\displaystyle \Omega =\ { (x,y)|x^ {2}+y^ {2}\leq 1\}} is the disk's surrounding circle: ∂ Ω = { ( x , y ) | x 2 + y 2 = 1 } General topology (Harrap, 1967). Lecture 2 Open Set and Interior Let X ⊆ Rn be a nonempty set Def. /pgf@ca0 << Selecting water in Figure 6 adds it to the project fluids section as the default fluid. The same area represented by a raster data model consists of several grid cells. As a consequence closed sets in the Zariski … /F31 18 0 R /ca 0.6 FIGURE 6. /Annots [ 56 0 R ] /Resources 60 0 R /Resources 63 0 R • The closure of A is the set c(A) := A∪d(A).This set is sometimes denoted by A. x��Z[oG~ϯط��x���(B���R��Hx0aV�M�4R|�ٙ��dl'i���Y��9���1��X����>��=x&X�%1ְ��2�R�gUu��:������{�Z}��ë�{��D1Yq�� �w+��Q J��t$���r�|�L����|��WBz������f5_�&F��A֯�X5��
�O����U�ăg�U�P�Z75�0g���DD �L��O�1r1?�/$�E��.F��j7x9a�n����$2C�����t+ƈ��y�Uf��|�ey��8?����/���L�R��q|��d�Ex�Ə����y�wǔ��Fa���a��lhE5�r`a��$� �#�[Qb��>����l�ش��J&:c_чpU��}�(������rC�ȱg�ӿf���5�A�s�MF��x%�#̧��Va�e�y�3�+�LITbq/�lkS��Q�?���>{8�2m��Ža$����EE�Vױ�-��RDF^�Z�RC������P Set Interior Closure Boundary f1g ? 3 min read. and also A [@A= Afor any set A. bwboundaries also descends into the outermost objects (parents) and traces their children (objects completely enclosed by the parents). /Pattern 15 0 R 9 The post office marks the [boundary] between the two municipalities. 26). � ¯ D = {(x, y) ∈ R2: x ≥ 0, y ≥ 0}. Coverings. >> /FontName /KLNYWQ+Cyklop-Regular 1 0 obj The set B is alsoa closed set. If fF Please Subscribe here, thank you!!! The set X is open if for every x ∈ X there is an open ball B(x,r) that entirely lies in the set X, i.e., for each x ∈ X there is r > 0 s.th. Interior, Closure, Exterior and Boundary Interior, Closure, Exterior and Boundary Example Let A = [0;1] [(2;3). A vector x0 is an interior point of the set X, if there is a ball B(x0,r) contained entirely in the set X Def. /CharSet (\057A\057B\057C\057E\057F\057G\057H\057I\057L\057M\057O\057P\057Q\057S\057T\057U\057a\057b\057bar\057c\057comma\057d\057e\057eight\057f\057ff\057fi\057five\057four\057g\057h\057hyphen\057i\057l\057m\057n\057nine\057o\057one\057p\057period\057r\057s\057seven\057six\057slash\057t\057three\057two\057u\057x\057y\057z\057zero) E X E R C IS E 1.1.1 . Interior and Boundary Points of a Set in a Metric Space Fold Unfold.