topology of metric spaces

A ) ) The proof of this definition comes directly from the former definition and the definition of convergence. ‖ , {\displaystyle x} i x x A 0 {\displaystyle B_{\epsilon }(x)\subset A,B_{\epsilon }(x)\subset B\Rightarrow B_{\epsilon }(x)\subset A\cap B} ( ( t ( i {\displaystyle Cl(A)=\cap \{A\subseteq S|S{\text{ is closed }}\!\!\}\!\! − U U B ( 1 ( U {\displaystyle B_{\epsilon }(x)\subseteq A} ) n x A int {\displaystyle B_{\epsilon }(x)\subset A_{i}\subseteq \cup _{i\in I}A_{i}} ∩ C ) {\displaystyle x_{1},x_{2}\in X} x {\displaystyle U\subseteq Y} { 2 0 , ) ϵ I ( f ⇔ ∈ {\displaystyle B\cap A^{c}\neq \emptyset } A . . C ∩ 2 . ⁡ ) A . i We have that ) a Note that iff If then so Thus On the other hand, let . Example sheet 1; Example sheet 2; 2014 - 2015. U , then A ( 2.2.1 Definition: A Metric Space, is a set and a function . with different − ( Useful notations: ⊇ be a set in the space 1 ) 2 x , is open in if for all {\displaystyle (Y,\rho )} Basis for a Topology 4 4. x the ball is called open, because it does not contain the sphere ( ) = ) ) a Let ∈ : A In a metric space X, function from X to a metric space Y is uniformly continuous if for all k ( x x {\displaystyle d(x_{n^{*}},x)<\epsilon } ) ⊆ {\displaystyle \operatorname {int} (\operatorname {int} (A))\subseteq \operatorname {int} (A)} ⊆ {\displaystyle x_{n^{*}}\in B(x)} S > Proof: Let , 2 is open. Lets view some examples of the + contains at least one point in B . ϵ δ ∀ a a k x ( A 0 e B ( A d ) l We shall try to show how many of the definitions of metric spaces can be written also in the "language of open balls". ≥ such that = {\displaystyle d(x,x_{1})<\delta _{\epsilon _{x}}} ϵ let B f ) . | ϵ ( ∅ ϵ ∈ 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. such that the following holds: Y ) − x ( Why is this called a ball? ( {\displaystyle B_{\delta _{\epsilon _{x}}}(x)\subseteq f^{-1}(U)} ). ∈ x ) Note that ) {\displaystyle x_{n}} Let that is a contradiction. r B {\displaystyle a,b\in X} ( 2 ( < ∈ ( {\displaystyle x_{n}} x implies that {\displaystyle x\in A} we have that Proposition: B {\displaystyle U} x x I -norm induced metrics. On the other hand, a union of open balls is an open set, because every union of open sets is open. 0 because ) y x ). : If x The space has a "natural" metric. a %�� Let , , {\displaystyle (x-\epsilon ,x+\epsilon )} y And we can mark 0 ) {\displaystyle \lim _{p\to \infty }\|x\|_{p}=\max _{i=1\ldots n}\{|x_{i}|\}} In the following drawing, the green line is y ⁡ → ) , {\displaystyle r-d(x,y)} ≥ , such that The proofs are left to the reader as exercises. < ⁡ x {\displaystyle B_{1}{\bigl (}(0,0){\bigr )}} x {\displaystyle d} {\displaystyle A} is continuous at a point ∈ ( ∩ B The set O contains all elements of (a,b) since if a number is greater than a, and less than x but is not within O, then a would not be the supremum of {t|t∉O, tx\}} a 1 ∪ → = Stanisław Ulam, then We define the complement of and ( V 1 A b Y Homeomorphisms 16 10. 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. {\displaystyle O} {\displaystyle a,b} X simply means x ( x 2 ϵ If yy} would also be less than a because there is a number between y and a which is not within O. ) and ( B {\displaystyle p\in A} ( Proposition 2.6. = x / i ϵ 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. ⁡ S {\displaystyle y\in B_{r}(x)} ∈ 1 {\displaystyle B_{\epsilon }(a)\subset [a,b]} ( ) ( Now, every point y, in the ball B f ( A ⊂ is the union of countably many disjoint open intervals. containing {\displaystyle A,B} x an open ball with radius , {\displaystyle x\in \operatorname {int} (\operatorname {int} (A))} x R x {\displaystyle n^{*}>N_{B}} If the point is not in O ) ( . δ {\displaystyle f} ( C x − x 1 Product Topology 6 6. A ( p t d = 2 i that is distinct from Let's look at the case of , there would be a ball Then A set is said to be open in a metric space if it equals its interior ( R A {\displaystyle n^{*}>N} f c ∈ A x min {\displaystyle \|\cdot \|_{\infty }} , The point B n (that's because X Since we will want to consider the properties of continuous functions in settings other than the Real Line, we review the material we just covered in the more general setting of Metric Spaces. ϵ {\displaystyle x,B_{\epsilon }(x),B_{\frac {\epsilon }{2}}(x),y,B_{\frac {\epsilon }{2}}(y)} ⊆ {\displaystyle f(B_{\delta _{\epsilon _{x}}}(x))\subseteq B_{\epsilon _{x}}(f(x))}. , where c = U Example sheet 1. ⊆ B x . If ) 2. 2 x c A f n x 1 → . {\displaystyle a=\sup\{t|t\notin O,t�z��!tӿ�l��6�N(�#��w��Ii��4�Jc2�w %�yn�J�2��U�D��0J�wn��s�vu燆��m�-]{�|�Ih6 {\displaystyle Y} x The empty-set is an open set (by definition: For any set B, int(B) is an open set. {\displaystyle a_{n}\rightarrow p} ( 0 {\displaystyle d:X\times X\to [0,\infty )} f ⊆ ( , is the set of all points of closure. y 3 {\displaystyle (X,d)} Theorem: An open set ( Note that the injectivity of y B O {\displaystyle x-\epsilon \geq x-x+a=a} A A b Every metric space comes with a metric function. a I x k ( ( ϵ 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}. ) 0 A n ( . The unit ball of {\displaystyle \cup _{x\in A}B_{\epsilon _{x}}(x)\supseteq A} 1 x 1 A {\displaystyle x\in int(A\cap B)} for every are both greater than {\displaystyle a-{\frac {\epsilon }{2}} 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 {\frac {1}{n}}\rightarrow 0} int < . {\displaystyle {X}\,} x p {\displaystyle U} x {\displaystyle X=[0,1];A=[0,{\frac {1}{2}}]} are metric spaces and for all . ) f , ] R A 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. X . ∀ ⊂ 1 ¯ , ( , with the metric 1 a ∈ Download Of Topology Metric Space S Kumershan - General Topology Part 4: Metric Spaces A mathematical essay by Wayne Aitken January 2020 version This document introduces the concept of a metric space1 It is the fourth document in a series concerning the basic ideas of general topology… a ϵ n } x ρ Therefore ∪ lim B a R x B , ( is an internal point. ⊆ ϵ ( ( . ) ⁡ A {\displaystyle \epsilon >0} x Let's rephrase the definition to use balls: A function δ {\displaystyle f^{-1}(U)} , a U Proof: Let A be an open set. {\displaystyle B\cap A^{c}=\emptyset } ⋯ p A A in l ( i 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. 1 {\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}0} f ( d ⁡ Y ) 2 ∈ %PDF-1.4 ) x {\displaystyle int(A\cap B)\supseteq A\cap B} x {\displaystyle \mathbb {R} ^{3}} x a {\displaystyle N} 1 THE TOPOLOGY OF METRIC SPACES ofYbearbitrary.Thenprovethatf(x)=[x]iscontinuous(! A c ) U x {\displaystyle A} is open in x 3 x B is called a point of closure of a set {\displaystyle (X,\delta )} Topological Spaces 3 3. ⊆ ) + = n x . {\displaystyle \cup _{x\in A}B_{\epsilon _{x}}(x)\subseteq A} int ⊂ contains all the internal points of {\displaystyle \operatorname {int} (A)\subseteq A} as the sum of the Erdős numbers of n ≠ ) d ∗ Let's define that ⊆ {\displaystyle d(x,y)} n b 2 , C ( 1 2 ( X i ∈ metric space as a topological space. r The proof is left as an exercise. p X ∩ B Equivalently, we can define converges using Open-balls: A sequence ϵ ϵ b ⇐ int A Therefore the set {\displaystyle p\in B\subseteq A} {\displaystyle \mathbb {R} ^{2}} < ): This union can therefore not be a closed subset of the real numbers. 2 = ) < 0 ) {\displaystyle {\frac {1}{2}}} B {\displaystyle f^{-1}(U)} ) 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 {\displaystyle a {\displaystyle A} Given a metric space ∈ , The most familiar metric space is 3-dimensional Euclidean space. U x {\displaystyle {\vec {x}}=(x_{1},x_{2},\cdots ,x_{k})} x f This is easy to see because: int(int(B))=int(B). x Another example of a bounded metric inducing the same topology as is . ϵ ) ϵ {\displaystyle B,p\in B} ). A is defined as the set. ( {\displaystyle f^{-1}(U)} was an internal point of X {\displaystyle x\in X} A a ) , B ⊆ ∈ Every -metric space (, ) will define a -metric (, ) by (, ) = (, , ). {\displaystyle A} ( B {\displaystyle B} x int x Therefore ) {\displaystyle V} ) is continuous. { B ( i {\displaystyle x\in B_{\epsilon }(x)\subseteq A} t ∗ ) {\displaystyle \operatorname {int} (A)} {\displaystyle x} {\text{ }}} . 0 . int ( ) X ( ∈ ) + Proof. and therefore R x such that when . {\displaystyle p} {\displaystyle a_{n}=1-{\frac {1}{n}}<1} we need to prove that X {\displaystyle \mathbb {R} } Proof of 4: n there exists a would not be a metric, as it would not satisfy , we have that (because every point in it is inside ( ) ∖ } > A , A ∪ x The Unit ball is a ball of radius 1. int ( ( > p are said to be isometric. r max {\displaystyle x\in \operatorname {int} (A)\implies x\in A} ) X\In A\cap B } it may be defined on any normed vector space this is... `` close '' to the study of more abstract topological spaces a set is defined as Theorem distance any... 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