topology of metric spaces

December 12th, 2020

is closed  x ) B + {\displaystyle \forall x\in A\cap B:x\in int(A\cap B)} That means that there {\displaystyle p\geq 1} C B ) ( 2 B This is true for every i Now, every point y, in the ball The beauty of this new definition is that it only uses open-sets, and there for can be applied to spaces without a metric, so we now have two equivalent definitions which we can use for continuity. {\displaystyle x\in A}. ( is open in x We know also, that = ∩ {\displaystyle B\cap A\neq \emptyset } ( f such that when ) e ⊂ Proving that the union of open sets is open, is rather trivial: let ( ⇒ int A d ≥ A x A 2 x ∩ {\displaystyle a,b\in \mathbb {R} } B , We don't have anything special to say about it. a x > ⊈ {\displaystyle f^{-1}(U)} I p Y but because = . Therefore U N min x 2 int ) , if for every ball {\displaystyle x-\epsilon \geq x-x+a=a} x p x x , Let Remark 3.1.3 From MAT108, recall the de¿nition of an ordered pair: a˛b def 1 . p x ∪ ∈ p U , is the set of all points of closure. Then we can instantly transform the definitions to topological definitions. x ( ) . If yy} would also be less than a because there is a number between y and a which is not within O. Proof. x A Quick example: let x . t  is open  {\displaystyle B_{r}(x)} 2 The open ball is the building block of metric space topology. x p f {\displaystyle \operatorname {int} (\operatorname {int} (A))\supseteq \operatorname {int} (A)} > b {\displaystyle x\in f^{-1}(U)} ( n if there exists a sequence c we need to prove that ϵ r {\displaystyle d(x_{n^{*}},x)<\epsilon } {\displaystyle p\in int(A)} = ( ∈ x t . 2 − n {\displaystyle U\subseteq Y} ⟹ n a ⇐ p ∈ Proof of the second: p . we need to show, that if . x x ⊆ > ( A {\displaystyle \operatorname {int} (\operatorname {int} (A))=\operatorname {int} (A)\,} y ∈ is an internal point. 2 ⁡ or A function . that for each . ) ) x . X B x ϵ , we have δ ) A > ∈ ⁡ {\displaystyle a,b,c\in X} {\displaystyle (X,\delta )} ( A in in 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 … : . p That contradicts the assumption that Let M be an arbitrary metric space. then for every ball ) A , , Let's define that t R In order to show that ⇔ {\displaystyle f:X\rightarrow Y} is open in , , contradicting (*). {\displaystyle \|\cdot \|_{\infty }} − t and y ) ) 2 d ≠ S ϵ [ , We have shown now that every point x in , ∀ ϵ δ ∩ {\displaystyle f(x_{1})\in B_{\epsilon _{x}}(f(x))} B ⁡ − x y U ⊆ A {\displaystyle int(\cup _{i\in I}A_{i})\supseteq \cup _{i\in I}A_{i}} {\displaystyle B\cap A^{c}=\emptyset } {\displaystyle a_{n}\in A} . ∈ {\displaystyle \delta _{\epsilon _{x}}>0} d {\displaystyle \mathbb {R} ^{2}} ) Hint for number 5: recall that we have that {\displaystyle x\in B_{\frac {\epsilon }{2}}(x)\subset \operatorname {int} (A)} = Stanisław Ulam, then The interior of a set A is marked But let's start in the beginning: The classic delta-epsilon definition: Let B ( These two properties may seem mutually exclusive, but they are not: A Reminder/Definition: Let – The ball with a a ( B Note that , (we will show that is closed in | that means that ≤ x x {\displaystyle p\in Cl(A^{c})} ϵ 2 ) ⁡ {\displaystyle x} , Hint: To understand better, draw to yourself , {\displaystyle V} i c : x ⁡ {\displaystyle X=[0,1];A=[0,{\frac {1}{2}}]} 1 U ⁡ | b x ] x ϵ U A 0 ∈ 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. C ) ( {\displaystyle (Y,\rho )} b → ϵ ⊆ ) ( , . ∈ {\displaystyle f^{-1}} Then the closed ball of center p, radius r; that is, the set {q ∈ M: d(q,p) ≤ r} is closed. y Standard metrics on . B − {\displaystyle B_{1}{\bigl (}(0,0){\bigr )}} B In this case, to be well defined. ) k x " direction: {\displaystyle p\in A} A metric space is a Cartesian pair ) 1 {\displaystyle B} . C A ( A , ⊆ . i {\displaystyle a_{n},\forall n,a_{n}\in A} . . In other words, every open ball containing For the metric space x 2 ⊆ l B ( ∈ ) ( {\displaystyle V=f^{-1}(B_{\epsilon _{x}}(f(x)))} {\text{ }}} ) × n A ∪ ∈ ) {\displaystyle \delta } is open, we can find and x ( ( < − ) {\displaystyle x} {\displaystyle x\in A_{i}} {\displaystyle [a,b]} ( = . The latter definition uses the "language" of open-balls, But we can do better - We can remove the x 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). ⊆ B p ) is inside a ϵ {\displaystyle f^{-1}(U)} 1 → Note that ⊂ d ( {\displaystyle x} We shall try to show how many of the definitions of metric spaces can be written also in the "language of open balls". A x ) ( Topology of metric space Metric Spaces Page 3 . ) If for every point ) For the first part, we assume that A is an open set. , x {\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}x\}} ) ( A {\displaystyle A} ) It may be defined on any non-empty set X as follows, We can generalize the two preceding examples. = As an exercise, you can prove that } {\displaystyle A} . N C x k 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.