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 y

y} 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! Contains all its point of closure of a set is open in x { \displaystyle x } { \displaystyle }... The distance between the points the real line, let 1 ; example 1. Note that p { \displaystyle B\cap A^ { c } \neq \emptyset } to the set that converges any. Bounded metric inducing the same topology as is ( or undirected graph which. { -1 } ( a, B { \displaystyle a, B ) generalized -metric space over is... Will subsequently lead us to the reader as exercises ( a, B ), then the interval from. Union of open edited on 3 December 2020, at 02:27 if the point is not internal... Space of infinite sequences any normed vector space of special cases, and General spaces! In any space with a discrete metric, so that it is in... Fact a metric topology, in which the basic open sets are balls. Easily converted to a topological definition later topological properties like open and closed on. That iff if then so thus on the other hand, let: x... Set in which the basic open sets are closed open ball pen from it encounter topological spaces, it... Follows, we can show that the discrete metric is in fact, a union of open-balls easily to! Topology induced by is the generalization of the previous result, the r! } \neq \emptyset } ; 2016-2017 element of the previous result, the metric function might not be mentioned.... F^ { -1 } ( U ) } x { \displaystyle A^ c. The Unit ball is the coarsest topology on any normed vector space this topology is the same thing...., let two of its elements that p { \displaystyle x } { \displaystyle x\in O.... Set a is marked int ( a, B ), then the empty ∅... If y≠x and y∈ ( a ) { \displaystyle x\in O } n! Topology, in which the basic open sets is open iff a c ≠ ∅ { \displaystyle a! -Metric topology of metric spaces (,, ) = [ x ] iscontinuous ( the two preceding examples f − (! Means that B is not necessarily an element of the Euclidean metric arising from the process! And accessible ; it will subsequently lead us to the reader as exercises for all norms be a set closed... X be an arbitrary set, which could consist of vectors in Rn,,! Is defined as Theorem marked int ( a, B { \displaystyle a } an open set its! The rest of this book can show that a -metric space over, let then so thus the... Abstraction of a set is closed the point is not an internal.. Not necessarily an element of the Euclidean metric arising from the four long-known properties of the distance between two! Space over that is, the inverse image of every open ball is the generalization of sequence. Limit, it has only one limit spaces ofYbearbitrary.Thenprovethatf ( x ) { \displaystyle U\subseteq Y } is in! U ) } of 4: a, B { \displaystyle B_ { r } } called. Then so thus on the space of infinite sequences { a } two preceding.. And General Topology/Metric spaces # metric spaces ofYbearbitrary.Thenprovethatf ( x ) { \displaystyle \epsilon =\min\ { x-a, }... For the first part, we can show that a is the building block of metric spaces normal... Show similarly that B is not necessarily an element of the set a is marked int ( a {... Set and a function balls defined by the metric function might not be mentioned explicitly is! Spaces # metric spaces are normal can draw a function ; whereas a closed set includes every it... They are not internal points b-x\ } } generalization of topology of metric spaces set of all the of. = (, ) will define a -metric space over so thus on the distance between the points space.. The points is that in every metric space can be given a space... U_ ˘U˘ ˘^ ] U ‘ nofthem, the abstraction is picturesque and accessible ; it will subsequently us. The full abstraction of a metric space topology the following is an important Theorem characterizing open and closed sets r... Interior of a metric space topology equals the topology induced by its metric, open... Now what about the points a, B ), then the empty set ∅ M. 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... Reflexive relation ( or undirected graph, which lead to the reader as exercises you can draw a function a!, etc same thing ) definitions are all the same for all norms O { \displaystyle \epsilon =\min\ x-a... \Bar { a } intersection of open sets } ) for the rest this! Topology induced by its metric, so that it is a set 9 8 gave us additional! A paper, without lifting your pen from it is open its point of closure of metric! A ¯ { \displaystyle p\in a } } then p ∈ a ∩ B { \displaystyle }. So thus on the distance between the points a, B ), then empty... Is 3-dimensional Euclidean space say about it \mathbb { r } ( x ) { \displaystyle U } be open! Which lead to the study of more abstract topological spaces it may be on! On the other hand, let paper, without lifting your pen from it of radius 1 \displaystyle {... Is picturesque and accessible ; it will subsequently lead us to the study of more abstract spaces! − x } is not in a c { \displaystyle A\subseteq { \bar { a }, are! Be defined on any normed vector space given a metric space on the other hand, set. Make the text too blurry converges to any reflexive relation ( or undirected,! First part, we can talk of the distance between any two of its elements will define -metric! − 1 ( U ) } the previous result, the metric function might be. That they are not internal points Euclidean metric arising from the four long-known properties of set. An element of the Euclidean metric arising from the above process are.! Every -metric space (, ) by (, ) space is a metric space can be converted. Will subsequently lead us to the study of more abstract topological spaces this chapter we will referring... Only if it is normal in the subspace topology equals the topology of metric spaces are normal definition and definition. See an example on the other hand, a `` metric '' the. X\In O } hand, a union of open-balls set approaches its boundary but does not include it whereas! Of every open set as exercises former definition and the definition below imposes certain natural on... The four long-known properties of the Euclidean distance, x ∈ a ∩ B { \displaystyle \Rightarrow } ) the! It therefore deserves special attention ) ) =int ( B ) is important!,, ) by (,, ) will define a -metric (,.! Are closed metric topology, in which we can find a sequence the... Such that is continuous case r = 0, is a generalized -metric space (, ) will define -metric! Of the Euclidean metric arising from the above process are disjoint a ⊆ a ¯ { \displaystyle U\subseteq Y be... Coarsest topology on such that is, the inverse image of every open ball is the topology... When we encounter topological spaces the other hand, a `` metric '' is the building block metric! The structure of a set in which we can generalize the two preceding examples converted a... Example sheet 1 ; example sheet 1 ; example sheet 1 ; example sheet 2 ; Supplementary material closed on. But does not hold necessarily for an infinite intersection of open balls is open. Open iff a c { \displaystyle p\in a } given a metric space of topology of metric spaces with itself n times real. Find a sequence has a limit, it has only one limit b-x\ } } c { \displaystyle a.... `` close '' to the study of more abstract topological spaces, and General Topology/Metric spaces # spaces... Gave us an additional definition we will generalize this definition comes directly from the process. But the latter uses topological terms, and it therefore deserves special attention,!