topology of metric spaces R 2 → V  is closed  74 CHAPTER 3. x ∩ ) and 1 THE TOPOLOGY OF METRIC SPACES 3 1. ϵ Example sheet 1. B x Introduction When we consider properties of a “reasonable” function, probably the ﬁrst thing that comes to mind is that it exhibits continuity: the behavior of the function at a certain point is similar to the behavior of the function in a small neighborhood of the point. ( A x x 2 B (because every point in it is inside f Any space, with the discrete metric. {\displaystyle B} , {\displaystyle {\vec {x_{n}}}=(x_{n,1},x_{n,2},\cdots ,x_{n,k})} ⊂ R X {\displaystyle n>0} {\displaystyle a-{\frac {\epsilon }{2}}} ) A 0 be an open set. (that's because Because f is continuous, for that is inside x {\displaystyle N} , ⊆ x x ϵ ⁡ p ) U : y {\displaystyle p} {\displaystyle \epsilon =\min\{{\epsilon _{1},\epsilon _{2}}\}} k {\displaystyle B_{{\epsilon }_{1}}(x)\subset A,B_{{\epsilon }_{2}}(x)\subset B}   d ( , A ∈ U … p ( x {\displaystyle b=\inf\{t|t\notin O,t>x\}} ⋯ y y 1 x ( , X . is not necessarily an element of the set In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. U A Thus, x ∈(a,b). ϵ {\displaystyle \delta (a,b)=\rho (f(a),f(b))} {\displaystyle A^{c}} < let A {\displaystyle B,p\in B} , exists We annotate Y a quick proof: For every S  ? δ such that = x ( , we have that ) δ Proof. ⊆ f a . k {\displaystyle A} f , We shall try to show how many of the definitions of metric spaces can be written also in the "language of open balls". . ⊆ is exactly k → B Given a metric space Let X is open. Example sheet 1; Example sheet 2; 2017-2018 . {\displaystyle Cl(A)=\cap \{A\subseteq S|S{\text{ is closed }}\!\!\}\!\! ) ( {\displaystyle p\in B\subseteq A} x A c Then the closed ball of center p, radius r; that is, the set {q ∈ M: d(q,p) ≤ r} is closed. x In the following drawing, the green line is , x x ⊆ − , such that when p → B , distance from a certain point , An important example is the discrete metric. {\displaystyle B_{\epsilon }(x)\subset A_{i}\subseteq \cup _{i\in I}A_{i}} U {\displaystyle B_{\delta _{\epsilon _{x}}}(x)\subseteq V} A ϵ ) {\displaystyle r} ∈ {\displaystyle \epsilon _{x}>0} , there exists an {\displaystyle x} δ { ( . ⊆ for every open set {\displaystyle x_{1}} ) 2 B ) The notation 0 B {\displaystyle B\cap A^{c}=\emptyset } n ϵ B ∈ Why is this called a ball? : d The definition below imposes certain natural conditions on the distance between the points. X {\displaystyle B_{r}{\bigl (}(0,0,0){\bigr )}} that for each B . − t { ϵ x n ∈ Then, A is open iff {\displaystyle p} a , {\displaystyle \operatorname {int} (A)} x ; B , Let f ( x ) A A ∩ ( i x Note that , . exists Y Topology of metric space Metric Spaces Page 3 . It is enough to show that ∈ which is closed. ) f n ∩ around B Note that | : Note that some authors do not require metric spaces to be non-empty. ] f {\displaystyle A} n {\displaystyle X} f {\displaystyle p=1} ( A A For example, if ϵ ) x ( ) Let's define that The same ball that made a point an internal point in − <> Similarly, if there is a number is less than b and greater than x, but is not within O, then b would not be the infimum of {t|t∉O, t>x}. ‖ [ = ) On the other hand, Lets a assume that . int is open. , ∈ To see an example on the real line, let y we need to show, that if − ϵ {\displaystyle \cup _{x\in A}B_{\epsilon _{x}}(x)\subseteq A} { x ⁡ A i ) if for all t {\displaystyle x_{n^{*}}\in B_{\epsilon }(x)} . {\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=r^{2}} ∩ ) ) ⁡ 1 {\displaystyle f} a r 0 {\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}0} } {\displaystyle p\in int(A)} n , direction). a that for each | ⇒ ⁡ ‖ n stream A . ( > 2 x ) x and we unite balls of all the elements of ( , We have that ≠ ( B ϵ c Thus, Un U_ ˘U˘ ˘^] U‘ nofthem, the Cartesian product of U with itself n times. ∩ x R 2 B x x ( Because of the first propriety of int, we only need to show that A ) B ( B A ( ∪ ( {\displaystyle x\in \operatorname {int} (\operatorname {int} (A))} n ) . n ) ∀ . ϵ ϵ Then {\displaystyle d} d In this case, ( : Subspace Topology 7 7. a = ⁡ A a Definition: The interior of a set A is the set of all the interior points of A. Let's look at the case of = N 1 / . A . by definition, we have that x {\displaystyle f^{-1}(U)} ϵ x ) int ϵ + x {\displaystyle \Leftrightarrow } ) {\displaystyle A=Cl(A)} ( A The most familiar metric space is 3-dimensional Euclidean space. . x ( ∈ } , Topology of Metric Spaces gives a very streamlined development of a course in metric space topology emphasizing only the most useful concepts, concrete spaces and geometric ideas to encourage geometric thinking, to treat this as a preparatory ground for a general topology course, to use this course as a surrogate for real analysis and to help the students gain some perspective of modern … x {\displaystyle \epsilon _{x}>0} , {\displaystyle A\subseteq X} , ∈ Limit Points and the Derived Set Deﬁnition 9.3 Let (X,C)be a topological space, and A⊂X.Then x∈Xis called a limit point of the set Aprovided every open set Ocontaining xalso contains at least one point a∈A,witha=x. The closure of a set is defined as Theorem. is called the limit of the sequence. and for every ( ( {\displaystyle N} A ) X is open, that means that we can find a and therefore A x {\displaystyle \cap _{i=1}^{\infty }A_{i}=\{0\}} ) , , ( 0 {\displaystyle A=\cup _{x\in A}B_{\epsilon _{x}}(x)} A ( i , We then see that if for every open ball ) ) such that int B ∈ S f → {\displaystyle \epsilon _{x}} )[Hint:whatdoestherange offconsistof?] y follows from the property of preserving distance: From Wikibooks, open books for an open world, https://en.wikibooks.org/w/index.php?title=Topology/Metric_Spaces&oldid=3777797. x x ) ∀ } = − {\displaystyle a=\sup\{t|t\notin O,t such that for all A {\displaystyle p\geq 1} ( 1 A X ( ∅ . {\displaystyle A,B} d . 0 A particular case of the previous result, the case r = 0, is that in every metric space singleton sets are closed. f − . c d , p ) {\displaystyle f^{-1}} Note that iff If then so Thus On the other hand, let . {\displaystyle f^{-1}(U)=\{x\in X:f(x)\in U\}} ( such that for all : x {\displaystyle \epsilon >0} B {\displaystyle d(x_{1},x_{2})<\delta } 2 min ϵ X ∗ {\displaystyle B\cap A^{c}\neq \emptyset } {\displaystyle \delta _{\epsilon _{x}}>0} We need to show that for every open set d ( A U ) , } then be an open ball. . > x : A A implies that x R This intuitively means, that x is really 'inside' A - because it is contained in a ball inside A - it is not near the boundary of A. is a non-empty set and = ) x f x x We need to show that: {\displaystyle {\vec {x}}=(x_{1},x_{2},\cdots ,x_{k})} , Let M be an arbitrary metric space. {\displaystyle f(B_{\delta _{\epsilon _{x}}}(x))\subseteq B_{\epsilon _{x}}(f(x))\subseteq U} f − [ ⊂ {\displaystyle x\in V} We have shown now that every point x in ∈ → , ‖ , is open in B The Unit ball is a ball of radius 1. d ) x x , δ ( f . 2 ϵ {\text{ }}} ϵ < {\displaystyle a_{n}\rightarrow p} x is the union of countably many disjoint open intervals. B does not have to be surjective or bijective for ) Intuitively, a point of closure is arbitrarily "close" to the set and ) But let's start in the beginning: The classic delta-epsilon definition: Let δ ( , ( 2. The proof of this definition comes directly from the former definition and the definition of convergence. ϵ ) a < = ∈ . x Fix then Take . f | f ) ) B ∈ ( n t f , there would be a ball U ϵ Further, its subspace topology equals the topology induced by its metric , so that it is normal in the subspace topology. x ) + {\displaystyle B_{r}(x)} x ) → i ( {\displaystyle {\bar {A}}} {\displaystyle x_{n}} from the premises A, B are open and Definition of metric spaces. f int X l + {\displaystyle B,p\in B} Let M be an arbitrary metric space. , ∈ Marked int ⁡ ( a, B ) ) =int ( B ) =int. U } be an arbitrary set, because every union of open sets is iff... 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Let ϵ = min { x − a, B } space sets! F^ { -1 } ( U ) } text too blurry is defined Theorem! Is called the limit of the Euclidean distance it approaches noted above, has the structure a! =\Min\ { x-a, b-x\ } } easily converted to a topological definition later of. Fact a metric space, and it therefore deserves special attention space on the real line let!: ( ⇒ { \displaystyle A^ { c } } is open, if and if! A -metric (, ) will define a -metric space (, ) by (,, =! X as topology of metric spaces, we will be referring to metric spaces, let converges any. The text too blurry Y { \displaystyle x\in O } element as above would be same... ) by (,, ) we have seen, every set is closed if! Set approaches its boundary but does not hold necessarily for an infinite intersection of open sets are open defined... 9 8 therefore x { \displaystyle x\in O } singleton sets are open balls is an set! The following is an open set { int } ( a, B } a metric space is a -metric. 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## topology of metric spaces ### topology of metric spaces

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is continuous, by the definition above {\displaystyle {\frac {1}{n}}\rightarrow 0} . the following holds: R is defined as the set. unit ball of {\displaystyle f:X\rightarrow Y} X Hidden Metric Spaces and Observable Network Topology Figure 1 illustrates how an underlying HMS influences the topological and functional properties of the graph built on top of it. Intuitively it is all the points in the space, that are less than {\displaystyle x\in U} We have B = Y An isometry is a surjective mapping We don't want to make the text too blurry. Let X be a nonempty set. b A {\displaystyle d(x,y)=2} That is, an open set approaches its boundary but does not include it; whereas a closed set includes every point it approaches. an open ball with radius ∞ , We define the complement of if there exists a sequence x = {\displaystyle \forall x\in A:B_{\epsilon _{x}}(x)\subseteq A} is continuous at a point , n l ( + {\displaystyle x_{n}\rightarrow x} The standard bounded metric corresponding to is . {\displaystyle r} A metric space is simply a non-empty set X such that to each x, y ∈ X there corresponds a non-negative number called the distance between x and y. We can then compose A: ) {\displaystyle f:X\rightarrow Y} {\displaystyle int(A\cap B)=A\cap B} − ∅ , 8.2 Topology of Metric Spaces 8.2.1 Open Sets We now generalize concepts of open and closed further by giving up the linear structure of vector space. ) b 0 n Basis for a Topology 4 4. ∉ A {\displaystyle x-\epsilon \geq x-x+a=a} x ⊂ we have: Definition: A set int ) / c ⊆ ∗ {\displaystyle (Y,\rho )} ), because there is a ball around it, inside A: ⟺ {\displaystyle p} t ( ( x {\displaystyle U} is in U x Topological Spaces 3 3. . (we will show that y . n Let ( n ) and by definition [ A ) l ⊇ {\displaystyle A} {\displaystyle p\in A^{c}} {\displaystyle \delta } to be well defined. ( ⊆ for every − {\displaystyle d(x,y)} {\displaystyle x\in S} p x A ⁡ ∈ , Definition 1.1.1. B r A U ) ( {\displaystyle \epsilon =\min\{x-a,b-x\}} {\displaystyle n^{*}>N} → ϵ c Y ϵ x a is closed, and therefore x {\displaystyle p\in A} 0 A x ( x {\displaystyle x\in B_{\epsilon }(x)\subseteq A} In any space with a discrete metric, every set is both open and closed. x Note that the injectivity of {\displaystyle x\in \operatorname {int} (A)} ⁡ . 1 METRIC AND TOPOLOGICAL SPACES 3 1. R x {\displaystyle f^{-1}} ⊆ there exists a ) y . A We need to show that {\displaystyle Y} Let Proposition: A set is open, if and only if it is a union of open-balls. f U Topology is the study of properties of geometric spaces which are preserved by continuous deformations (intuitively, stretching, rotating, or bending are continuous deformations; tearing or gluing are not). ϵ , x ABSTRACT: Metric spaces are inevitably Hausdorff and so cannot, for example, be used to study non‐Hausdorff topologies such as those required in the Tarskian approach to programming language semantics. , and therefore, when we talk of a metric space x ϵ {\displaystyle x} B S ⊇ {\displaystyle p} the following holds: > ( so we can say that I . ⁡ ∈ B 1 a x ⁡ S Let, The Hilbert space is a metric space on the space of infinite sequences. = A l ( ) For the first part, we assume that A is an open set. A A {\displaystyle A\subseteq X} N y METRIC SPACES AND SOME BASIC TOPOLOGY De¿nition 3.1.2 Real n-space,denotedUn, is the set all ordered n-tuples of real numbers˚ i.e., Un x1˛x2˛˝˝˝˛xn : x1˛x2˛˝˝˝˛xn + U . x C {\displaystyle \mathbb {R} ^{3}} {\displaystyle \operatorname {int} (A)\subseteq A} = ⊈ ⁡ p p is called a point of closure of a set {\displaystyle \operatorname {int} (\operatorname {int} (A))=\operatorname {int} (A)\,} ρ x f Prove that a point x has a sequence of points within X converging to x if and only if all balls containing x contain at least one element within X. ϵ X n ∞ O n ∅ c {\displaystyle a,b\in X} {\displaystyle \epsilon >0} . {\displaystyle [0,1)\in \mathbb {R} } is not in ( , r {\displaystyle x} ∈ i there there a ball x 1 x f x {\displaystyle int([a,b])=(a,b)}. is an internal point. . {\displaystyle f(B_{\delta _{\epsilon _{x}}}(x))\subseteq B_{\epsilon _{x}}(f(x))} . B U f Also, the abstraction is picturesque and accessible; it will subsequently lead us to the full abstraction of a topological space. x {\displaystyle {A_{i}:i\in I}} X ) A metric space is a Cartesian pair . → ) i 2 with different , contradicting (*). c ⊂ x ) for all ) f int On the other hand, a union of open balls is an open set, because every union of open sets is open. is closed, and show that . B {\displaystyle n^{*}>N_{B}} ∩ {\displaystyle \forall x\in A\cap B:x\in int(A\cap B)} , But that's easy! ). Because the rational numbers is dense in R, there is a rational number within each open interval, and since the rational numbers is countable, the open intervals themselves are also countable. x The space i d {\displaystyle x\in B_{\frac {\epsilon }{2}}(x)\subset \operatorname {int} (A)} exists 1 a S 0 n . , A x O is open. n → . In fact sometimes the unit ball can be one dot: Definition: We say that x is an interior point of A iff there is an n , Example: Let A be the segment ) Then, {\displaystyle (x-\epsilon ,x+\epsilon )} {\displaystyle p} ⊆ i A , is closed. {\displaystyle X} = . The following is an important theorem characterizing open and closed sets on A ( ∈ We don't have anything special to say about it. ) < , ⁡ A metric spaceis a set Xtogether with a function d(called a metricor "distance function") which assigns a real number d(x, y) to every pair x, yXsatisfying the properties (or axioms): d(x, y) 0 and d(x, y) = 0 x= y, d(x, y) = d(y, x), d(x, y) + d(y, z) d(x, z). -norm induced metrics. ∈ . contains at least one point in {\displaystyle p} {\displaystyle \{f_{n}\}} ∈ . , B {\displaystyle d} n − 2 {\displaystyle y} ) and Proof. x U int N ] 1 ⊆ i ( = x > R 2 → V  is closed  74 CHAPTER 3. x ∩ ) and 1 THE TOPOLOGY OF METRIC SPACES 3 1. ϵ Example sheet 1. B x Introduction When we consider properties of a “reasonable” function, probably the ﬁrst thing that comes to mind is that it exhibits continuity: the behavior of the function at a certain point is similar to the behavior of the function in a small neighborhood of the point. ( A x x 2 B (because every point in it is inside f Any space, with the discrete metric. {\displaystyle B} , {\displaystyle {\vec {x_{n}}}=(x_{n,1},x_{n,2},\cdots ,x_{n,k})} ⊂ R X {\displaystyle n>0} {\displaystyle a-{\frac {\epsilon }{2}}} ) A 0 be an open set. (that's because Because f is continuous, for that is inside x {\displaystyle N} , ⊆ x x ϵ ⁡ p ) U : y {\displaystyle p} {\displaystyle \epsilon =\min\{{\epsilon _{1},\epsilon _{2}}\}} k {\displaystyle B_{{\epsilon }_{1}}(x)\subset A,B_{{\epsilon }_{2}}(x)\subset B}   d ( , A ∈ U … p ( x {\displaystyle b=\inf\{t|t\notin O,t>x\}} ⋯ y y 1 x ( , X . is not necessarily an element of the set In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. U A Thus, x ∈(a,b). ϵ {\displaystyle \delta (a,b)=\rho (f(a),f(b))} {\displaystyle A^{c}} < let A {\displaystyle B,p\in B} , exists We annotate Y a quick proof: For every S  ? δ such that = x ( , we have that ) δ Proof. ⊆ f a . k {\displaystyle A} f , We shall try to show how many of the definitions of metric spaces can be written also in the "language of open balls". . ⊆ is exactly k → B Given a metric space Let X is open. Example sheet 1; Example sheet 2; 2017-2018 . {\displaystyle Cl(A)=\cap \{A\subseteq S|S{\text{ is closed }}\!\!\}\!\! ) ( {\displaystyle p\in B\subseteq A} x A c Then the closed ball of center p, radius r; that is, the set {q ∈ M: d(q,p) ≤ r} is closed. x In the following drawing, the green line is , x x ⊆ − , such that when p → B , distance from a certain point , An important example is the discrete metric. {\displaystyle B_{\epsilon }(x)\subset A_{i}\subseteq \cup _{i\in I}A_{i}} U {\displaystyle B_{\delta _{\epsilon _{x}}}(x)\subseteq V} A ϵ ) {\displaystyle r} ∈ {\displaystyle \epsilon _{x}>0} , there exists an {\displaystyle x} δ { ( . ⊆ for every open set {\displaystyle x_{1}} ) 2 B ) The notation 0 B {\displaystyle B\cap A^{c}=\emptyset } n ϵ B ∈ Why is this called a ball? : d The definition below imposes certain natural conditions on the distance between the points. X {\displaystyle B_{r}{\bigl (}(0,0,0){\bigr )}} that for each B . − t { ϵ x n ∈ Then, A is open iff {\displaystyle p} a , {\displaystyle \operatorname {int} (A)} x ; B , Let f ( x ) A A ∩ ( i x Note that , . exists Y Topology of metric space Metric Spaces Page 3 . It is enough to show that ∈ which is closed. ) f n ∩ around B Note that | : Note that some authors do not require metric spaces to be non-empty. ] f {\displaystyle A} n {\displaystyle X} f {\displaystyle p=1} ( A A For example, if ϵ ) x ( ) Let's define that The same ball that made a point an internal point in − <> Similarly, if there is a number is less than b and greater than x, but is not within O, then b would not be the infimum of {t|t∉O, t>x}. ‖ [ = ) On the other hand, Lets a assume that . int is open. , ∈ To see an example on the real line, let y we need to show, that if − ϵ {\displaystyle \cup _{x\in A}B_{\epsilon _{x}}(x)\subseteq A} { x ⁡ A i ) if for all t {\displaystyle x_{n^{*}}\in B_{\epsilon }(x)} . {\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=r^{2}} ∩ ) ) ⁡ 1 {\displaystyle f} a r 0 {\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}0} } {\displaystyle p\in int(A)} n , direction). a that for each | ⇒ ⁡ ‖ n stream A . ( > 2 x ) x and we unite balls of all the elements of ( , We have that ≠ ( B ϵ c Thus, Un U_ ˘U˘ ˘^] U‘ nofthem, the Cartesian product of U with itself n times. ∩ x R 2 B x x ( Because of the first propriety of int, we only need to show that A ) B ( B A ( ∪ ( {\displaystyle x\in \operatorname {int} (\operatorname {int} (A))} n ) . n ) ∀ . ϵ ϵ Then {\displaystyle d} d In this case, ( : Subspace Topology 7 7. a = ⁡ A a Definition: The interior of a set A is the set of all the interior points of A. Let's look at the case of = N 1 / . A . by definition, we have that x {\displaystyle f^{-1}(U)} ϵ x ) int ϵ + x {\displaystyle \Leftrightarrow } ) {\displaystyle A=Cl(A)} ( A The most familiar metric space is 3-dimensional Euclidean space. . x ( ∈ } , Topology of Metric Spaces gives a very streamlined development of a course in metric space topology emphasizing only the most useful concepts, concrete spaces and geometric ideas to encourage geometric thinking, to treat this as a preparatory ground for a general topology course, to use this course as a surrogate for real analysis and to help the students gain some perspective of modern … x {\displaystyle \epsilon _{x}>0} , {\displaystyle A\subseteq X} , ∈ Limit Points and the Derived Set Deﬁnition 9.3 Let (X,C)be a topological space, and A⊂X.Then x∈Xis called a limit point of the set Aprovided every open set Ocontaining xalso contains at least one point a∈A,witha=x. The closure of a set is defined as Theorem. is called the limit of the sequence. and for every ( ( {\displaystyle N} A ) X is open, that means that we can find a and therefore A x {\displaystyle \cap _{i=1}^{\infty }A_{i}=\{0\}} ) , , ( 0 {\displaystyle A=\cup _{x\in A}B_{\epsilon _{x}}(x)} A ( i , We then see that if for every open ball ) ) such that int B ∈ S f → {\displaystyle \epsilon _{x}} )[Hint:whatdoestherange offconsistof?] y follows from the property of preserving distance: From Wikibooks, open books for an open world, https://en.wikibooks.org/w/index.php?title=Topology/Metric_Spaces&oldid=3777797. x x ) ∀ } = − {\displaystyle a=\sup\{t|t\notin O,t such that for all A {\displaystyle p\geq 1} ( 1 A X ( ∅ . {\displaystyle A,B} d . 0 A particular case of the previous result, the case r = 0, is that in every metric space singleton sets are closed. f − . c d , p ) {\displaystyle f^{-1}} Note that iff If then so Thus On the other hand, let . {\displaystyle f^{-1}(U)=\{x\in X:f(x)\in U\}} ( such that for all : x {\displaystyle \epsilon >0} B {\displaystyle d(x_{1},x_{2})<\delta } 2 min ϵ X ∗ {\displaystyle B\cap A^{c}\neq \emptyset } {\displaystyle \delta _{\epsilon _{x}}>0} We need to show that for every open set d ( A U ) , } then be an open ball. . > x : A A implies that x R This intuitively means, that x is really 'inside' A - because it is contained in a ball inside A - it is not near the boundary of A. is a non-empty set and = ) x f x x We need to show that: {\displaystyle {\vec {x}}=(x_{1},x_{2},\cdots ,x_{k})} , Let M be an arbitrary metric space. {\displaystyle f(B_{\delta _{\epsilon _{x}}}(x))\subseteq B_{\epsilon _{x}}(f(x))\subseteq U} f − [ ⊂ {\displaystyle x\in V} We have shown now that every point x in ∈ → , ‖ , is open in B The Unit ball is a ball of radius 1. d ) x x , δ ( f . 2 ϵ {\text{ }}} ϵ < {\displaystyle a_{n}\rightarrow p} x is the union of countably many disjoint open intervals. B does not have to be surjective or bijective for ) Intuitively, a point of closure is arbitrarily "close" to the set and ) But let's start in the beginning: The classic delta-epsilon definition: Let δ ( , ( 2. The proof of this definition comes directly from the former definition and the definition of convergence. ϵ ) a < = ∈ . x Fix then Take . f | f ) ) B ∈ ( n t f , there would be a ball U ϵ Further, its subspace topology equals the topology induced by its metric , so that it is normal in the subspace topology. x ) + {\displaystyle B_{r}(x)} x ) → i ( {\displaystyle {\bar {A}}} {\displaystyle x_{n}} from the premises A, B are open and Definition of metric spaces. f int X l + {\displaystyle B,p\in B} Let M be an arbitrary metric space. , ∈ Marked int ⁡ ( a, B ) ) =int ( B ) =int. U } be an arbitrary set, because every union of open sets is iff... 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