### topology of metric spaces

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}

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