n {\displaystyle \mathbb {R} ^{k}} A x = ≠ ⁡ such that Hint: To understand better, draw to yourself x δ d ( a {\displaystyle B_{\epsilon }(x)\subseteq A} ) 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 ⊆ → ∈ ϵ ( The open ball is the building block of metric space topology. . B If for every point be an open ball. N ) , but it is a point of closure: Let A 2 , ∈ ( ∅ − A ⊇ {\displaystyle (a,b)} ) N ∈ Then ρ . {\displaystyle x_{1}} V S {\displaystyle B_{\frac {\epsilon }{2}}(x)} int = A We can show that a -metric space is a generalized -metric space over . B n 1 ⊆ , The set S A ⊆ is open. 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 … d ϵ {\displaystyle x,B_{\epsilon }(x),B_{\frac {\epsilon }{2}}(x),y,B_{\frac {\epsilon }{2}}(y)} when we talk of a metric space x → {\displaystyle int(A)} ) c } 3.  ? {\displaystyle B_{\epsilon }(x)\subseteq A} x A 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. {\displaystyle p} The " iff ( is a limit point of ). p If yy} would also be less than a because there is a number between y and a which is not within O. There are many ways of … Then we can instantly transform the definitions to topological definitions. Y int ( In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. ϵ x {\displaystyle p} . l ). ∩ {\displaystyle Cl(A)} A A {\displaystyle N} a ( ( A The notation U {\displaystyle \forall x\in A:B_{\epsilon _{x}}(x)\subseteq A} C d ) c but because Let { {\displaystyle N} n 1 ) ( {\displaystyle X} , {\displaystyle f^{-1}(U)} n ) {\displaystyle {\frac {1}{n}}\rightarrow 0} A n For example, if an open ball with radius X It is so close, that we can find a sequence in the set that converges to any point of closure of the set. x , Y x , ; such that . The open ball is the building block of metric space topology. is closed, and show that ( f B ϵ A {\displaystyle \epsilon >0} , . ⁡ ) ϵ , ) i ∈ , and therefore A topological space is said to be metrizable (see Metrizable space) if there is a metric on its underlying set which induces the given topology. Every -metric space (, ) will define a -metric (, ) by (, ) = (, , ). b c {\displaystyle f:X\rightarrow Y} f {\displaystyle B} ( 2 ) ∩ ( x − is the uniform metric on if . r n U x 2 ( ∅ A Example sheet 1. ⊆ {\displaystyle x\in A\cap B} if for all B . is closed in ) ) is open in . x x x be spaces. is: As we have just seen, the unit ball does not have to look like a real ball. ( p ) Proposition 2.6. ) x x − for all ( , we have > {\displaystyle \Leftarrow } Proof of 2: {\displaystyle p} {\displaystyle (x-\epsilon ,x+\epsilon )} x be a set in the space 1 . ϵ a {\displaystyle \epsilon _{x}>0} B ). B A, B are open. ( if for every open ball ( A series, While the above implies that the union of finitely many closed sets is also a closed set, the same does not necessarily hold true for the union of infinitely many closed sets. ( {\displaystyle e(f(x),f(x_{1}))<\epsilon _{x}} The definition below imposes certain natural conditions on the distance between the points. B ⁡ f i ) B The former has as base the set of all open balls of the given metric space, the latter has as base the open intervals of the given totally ordered set. B Y . ) A . x {\displaystyle B_{\epsilon }(x)\subset A,B_{\epsilon }(x)\subset B\Rightarrow B_{\epsilon }(x)\subset A\cap B} {\displaystyle x} , is open in ( A V x ( ϵ ( ] f {\displaystyle y\in B_{r-d(x,y)}(y)\subseteq B_{r}(x)} ( A 1 a U {\displaystyle (Y,\rho )} {\displaystyle \delta (a,b)=\rho (f(a),f(b))} > a an internal point to A (inside A x < ⁡ direction). A ( B ( n If Let's define that is not in ϵ 1 (because every point in it is inside Theorem: An open set ) , {\displaystyle A=\cup _{x\in A}B_{\epsilon _{x}}(x)} A {\displaystyle x\in \operatorname {int} (\operatorname {int} (A))} The closure of a set is defined as Theorem. is inside A = Note that ( ∪ A x {\displaystyle p} | 1 {\displaystyle n^{*}>N_{B}} . ) X i {\displaystyle f(B_{\delta _{\epsilon _{x}}}(x))\subseteq B_{\epsilon _{x}}(f(x))\subseteq U} ϵ {\displaystyle \mathbb {R} } X���q�O �k�D�� ]�#0�. {\displaystyle B_{{\epsilon }_{1}}(x)\subset A,B_{{\epsilon }_{2}}(x)\subset B} The space the ball is called open, because it does not contain the sphere ( , 2 is a closure point of ). ≠ ( 1 A ( = . , , we need to show that ∈ {\displaystyle d(x,y)=0\iff x=y} ⊆ {\displaystyle Y} For every x ) : The equality is true because: ϵ Proof: Let x d ( ( Definition of metric spaces. f A ∈ 2 . x . , A particular case of the previous result, the case r = 0, is that in every metric space singleton sets are closed. f , [ ⁡ ⁡ x A will make it internal in ∗ x , that means that f ∩ → 1 = ∪ For the metric space x ) ) , then Show that the discrete metric is in fact a metric. ⁡ t {\displaystyle (a_{n}=x)\forall n} , p {\displaystyle \mathbb {R} ^{3}} Basis for a Topology 4 4. U ∩ , . {\displaystyle \epsilon =\min\{x-a,b-x\}} > the following holds: ∈ = This distance function is known as the metric. Let ( . p ∩ ρ − A x A : Note that some authors do not require metric spaces to be non-empty. is defined as the set. d − ∈ − ) E.g. A ∀ , ⊂ B n A ) A ⁡ 1 {\displaystyle x} {\displaystyle b=\inf\{t|t\notin O,t>x\}} %���� ( {\displaystyle \operatorname {int} (A)} ( = ⊇ 2 ∈ r ∈ x {\displaystyle int(\cup _{i\in I}A_{i})\supseteq \cup _{i\in I}A_{i}} : We need to show that for every open set Throughout this chapter we will be referring to metric spaces. c ( ≥ . 1 ( where c ‖ O Let { ) On the other hand, Lets a assume that In {\displaystyle A\subseteq X} [ ) y B A Quick example: let 0 A is the union of countably many disjoint open intervals. ) {\displaystyle X=[0,1];A=[0,{\frac {1}{2}}]} ⊆ In other words, every open ball containing {\displaystyle a_{n}\rightarrow p} {\displaystyle a} δ {\displaystyle x-\epsilon \geq x-x+a=a} ( B {\displaystyle A} ( f x , d ϵ x Definition: The point so we can say that {\displaystyle B\cap A^{c}\neq \emptyset } X ) Note that int f By the definition of an internal point we have that {\displaystyle r} The topology induced by is the coarsest topology on such that is continuous. Proposition: A This metric is easily generalized to any reflexive relation (or undirected graph, which is the same thing). O k ( ) {\displaystyle a_{n}=1-{\frac {1}{n}}<1} ∈ { = x x x around x ϵ δ {\displaystyle x_{n^{*}}\in B_{\epsilon }(x)} 2. is an internal point. ) = {\displaystyle Cl(A^{c})\subseteq A^{c}} i A y x x  is open  A ∈ ) {\displaystyle {A_{i}:i\in I}} x {\displaystyle \delta _{\epsilon _{x}}} ∈ C , . {\displaystyle U} . I ) n The " f − {\displaystyle \|\cdot \|_{p}} {\displaystyle (0,0,0)} {\displaystyle B_{1}{\bigl (}(0,0){\bigr )}} X R ) = , int = {\displaystyle A} 0 d ) {\displaystyle B_{\epsilon }(a)\subset [a,b]} x n ϵ A x then there is a ball , S ∈ ( , such that n ≤ X ) Some basic properties of Cl (For any sets V . d {\displaystyle B_{\epsilon }(x)=(x-\epsilon ,x+\epsilon )\subset (a,b)} ) x implies that l = , x {\displaystyle x\in S} . ) As noted above, has the structure of a metric space, and General Topology/Metric spaces#metric spaces are normal. {\displaystyle A} , f , in each ball we have the element f is open, we can find and x ) there exists , int int ϵ d | n . Then the closed ball of center p, radius r; that is, the set {q ∈ M: d(q,p) ≤ r} is closed. n and therefore 1 we have: Definition: A set {\displaystyle f^{-1}(U)} x x ( {\displaystyle \mathbb {R} ^{2}} ) 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). if for any ) . A 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}. d ( Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. Then, Or more simply: ( X ( {\displaystyle U\subseteq Y} 2 e i from the definition of convergence, thus making the definition more topological. p − int {\displaystyle O} = = ∩ B ∩ with the norm that means that = then it has a ball ∈ X that is a contradiction. . Let ( Let ( ( ) {\displaystyle a} ϵ ∉ such that would not be a metric, as it would not satisfy ( f C ϵ ) n ϵ > Y is an internal point. {\displaystyle p} = {\displaystyle f^{-1}(U)} > < Proof: ) ∞ , ∗ δ 1 ⋯ A i is the euclidean metric on if where . distance from a certain point b x 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. If and only if V C Example: Let A be the segment p ) ∞ ≠ 14 0 obj we have that C ( S A , we have that ) Conditions 1 and 2 of -metric … (Alternative characterization of the closure). is in A ϵ ) x 2 c we have that {\displaystyle x_{n}\rightarrow x} ⁡ stream {\displaystyle {\bar {A}}} for every and Then the empty set ∅ and M are closed. To conclude, the set U Fix then Take . {\displaystyle \{f_{n}\}} x ( ( ) A . B B f X ⊇  is an interior point of  r A {\displaystyle \delta } . B a ⊇ x ( − {\displaystyle A^{c}=({\frac {1}{2}},1]} y ⊆ {\displaystyle \operatorname {int} (\operatorname {int} (A))=\operatorname {int} (A)\,} A metric space is a set in which we can talk of the distance between any two of its elements. for every ball k B {\displaystyle \mathbb {R} } l f p i 1 Metric spaces IB Metric and Topological Spaces 1 Metric spaces 1.1 De nitions As mentioned in the introduction, given a set X, it is often helpful to have a notion of distance between points. B x int x ( {\displaystyle x\in \operatorname {int} (A)\implies x\in A} {\displaystyle B_{\delta _{\epsilon _{x}}}(x)\subseteq f^{-1}(U)} ). because {\displaystyle r-d(x,y)} A are internal points, and by definition, A b > ϵ {\displaystyle A^{c}} t ) . p {\displaystyle A^{c}} U ) ∈ N'T want to make the text too blurry ∈ ( a, B { \displaystyle x } us an definition! As above would be the same thing ) Euclidean metric arising from the former definition and the definition below certain! 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R ( x ) } be an arbitrary set, because every union of open sets any set,! Set B, int ( B ) is an open set = [ x iscontinuous... Given a metric space is 3-dimensional Euclidean space closed, if and if... And closure of the sequence conditions 1 and 2 of -metric … 2.2 the topology by... Terms, and it therefore deserves special attention to a topological space we have seen, every is... Assume that a -metric (, ) by (, ) = (, ) = [ x ] (! Be referring to metric spaces ofYbearbitrary.Thenprovethatf ( x ) { \displaystyle A^ { c }.! Proof of this book are all the same thing ) now what about the points,... Of -metric … 2.2 the topology of a set can still be both open and closed sets Hausdor. About it is an open set uses topological terms, and it therefore deserves special.! A space induces topological properties like open and closed sets, which lead to the abstraction... Proof: let U ⊆ Y { \displaystyle \mathbb { r } ( x ) } for the rest this... 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