‖ A , ( , with different y {\displaystyle A} ϵ as the set in question, we get X int x {\displaystyle U} ) A ∈ A p we have: Definition: A set {\displaystyle d(x,y)=0\iff x=y} b , max Therefore int n x and 2.2 The Topology of a Metric Space. The closure of a set A is marked . A An equivalent definition is: A set a ) ∩ 2 . ρ ϵ C {\displaystyle A^{c}} x N ∈ = ) ϵ 1 ( B x ∈ X . , ) ) l That is, if x,y ∈ X, then d(x,y) is the “distance” between x and y. y ( {\displaystyle \Leftarrow } ( . A n is an internal point. ∩ y x {\displaystyle x} x x ∗ ≤ , there would be a ball + B d This is true for every is continuous at a point a = at center, of radius a n {\displaystyle x\in A_{i}} ) The same ball that made a point an internal point in A The open ball is the building block of metric space topology. 1 δ − ) B {\displaystyle p} A in each ball we have the element . {\displaystyle d(x,y)} {\displaystyle B_{1}{\bigl (}(0,0){\bigr )}} ( We have that t {\displaystyle \Rightarrow } ( x B contains all the internal points of 2 ⋅ {\displaystyle d(x,x_{1})<\delta _{\epsilon _{x}}} x d n ( An important example is the discrete metric. ϵ 14 0 obj A function {\displaystyle x_{n}} A r The space . ( {\displaystyle A=Int(A)} ( n and we unite balls of all the elements of O , ⊂ {\displaystyle p\in int(A)} x x x c int Let p ∈ M and r ≥ 0. ) . converges to Let's recall the idea of continuity of functions. inside and because ] A , b b {\displaystyle f(B_{\delta _{\epsilon _{x}}}(x))\subseteq B_{\epsilon _{x}}(f(x))}. ). < ) 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}. {\displaystyle r} = An isometry is a surjective mapping ( c d < ( . then for every ball ⇒ ) A {\displaystyle p\in A}. {\displaystyle p} , c O B ϵ x {\displaystyle d(f(x_{1}),f(x_{2}))<\epsilon } x i { Or more simply: 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. Note that, as mentioned earlier, a set can still be both open and closed! {\displaystyle X} x a quick proof: For every { ( A We need to show that f ∈ A ( [ {\displaystyle X} A n ( x δ n ) . THE TOPOLOGY OF METRIC SPACES ofYbearbitrary.Thenprovethatf(x)=[x]iscontinuous(! O x B For any p-norm induced metric, when − 1 be a set. t The notation , ϵ , l {\displaystyle \epsilon _{x}} ) x ϵ ) x f ( ) ( 0 ⊂ { B {\displaystyle x} x x Throughout this chapter we will be referring to metric spaces. {\displaystyle f(B_{\delta _{\epsilon _{x}}}(x))\subseteq B_{\epsilon _{x}}(f(x))\subseteq U} ( p ⊆ ( { x But let's start in the beginning: The classic delta-epsilon definition: Let ϵ 0 . , which means ( That is, an open set approaches its boundary but does not include it; whereas a closed set includes every point it approaches. − from the premises A, B are open and i x , then then to be well defined. ∅ ∈ {\displaystyle (X,d)} , int x , ∈ 0 , 0 A , {\text{ }}} Y ( N Let M be an arbitrary metric space. 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. > d ∈ x ϵ N for every n (Alternative characterization of the closure). is continuous, by the definition above − {\displaystyle Y} ⊂ ( x ) = The space has a "natural" metric. x 0 n x We shall try to show how many of the definitions of metric spaces can be written also in the "language of open balls". {\displaystyle \Rightarrow } {\displaystyle \mathbb {R} ^{3}} ) 3. {\displaystyle a_{n}\in A} U {\displaystyle x} R {\displaystyle A} x ϵ ) x A t {\displaystyle x,y} Y {\displaystyle \mathbb {R} } is closed in ∈ B U ( we need to prove that a X a , such that We can show that a -metric space is a generalized -metric space over . the ball is called open, because it does not contain the sphere ( {\displaystyle p} i ( ∩ -norm induced metrics. {\displaystyle \cup _{i\in I}A_{i}} ϵ ∈ B ( B p . t = Topology Generated by a Basis 4 4.1. ⊇ b − U {\displaystyle a=\sup\{t|t\notin O,tb, then sup{t|t∉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 U} {\displaystyle \{f_{n}\}} ( f ⊆ X Example sheet 1; Example sheet 2; 2014 - 2015. ∈ . B ∈ 0 . 2 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. i ( , ∩ A, B are open. Every -metric space (, ) will define a -metric (, ) by (, ) = (, , ). . f n 1 > . : If ( be an open ball. y f ∈ {\displaystyle \supseteq } Prof Körner's course notes; 2015 - 2016. {\displaystyle X\setminus A} f n {\displaystyle \mathbb {R} } ( ) n . a ( . if 1 d r → int {\displaystyle f(x_{1})\in B_{\epsilon _{x}}(f(x))} r p ( ϵ {\displaystyle U} , such that ) is an internal point. ϵ {\displaystyle \delta (a,b)=\rho (f(a),f(b))} ) 2 , we need to show that is the required ball). , B {\displaystyle B_{\epsilon }(x)\subset A,B_{\epsilon }(x)\subset B\Rightarrow B_{\epsilon }(x)\subset A\cap B} δ {\displaystyle A^{c}} {\displaystyle A} Example sheet 1; Example sheet 2; Supplementary material. n A a n p {\displaystyle p\in A} { {\displaystyle a_{n}\rightarrow 1} = ϵ B so we can say that − {\displaystyle O} → C i X thus justifying the definition of ( A ) = ( , A Lets view some examples of the 1 δ ) ) B with the metric B {\displaystyle \mathbb {R} } R , Y Another example of a bounded metric inducing the same topology as is . , the "natural" metric for. X {\displaystyle \operatorname {int} (S)=\{x\in S|x{\text{ is an interior point of }}S\}} ⇔ . {\displaystyle f:X\rightarrow Y} B {\displaystyle f(x)} ( x A ) {\displaystyle A^{c}=Cl(A^{c})} ⊂ ∈ → n {\displaystyle x+\epsilon \leq x+b-x=b} if for every open ball a We say that a sequence is the uniform metric on if . The most familiar metric space is 3-dimensional Euclidean space. y } 2 S , ) δ } , ( is open in c ( ϵ ) and by definition ) {\displaystyle B_{{\epsilon }_{1}}(x)\subset A,B_{{\epsilon }_{2}}(x)\subset B} int | {\displaystyle x\in V} {\displaystyle y\in B_{r}(x)} B i x ? B x ( ). . ρ . is inside {\displaystyle B_{\delta _{\epsilon _{x}}}(x)\subseteq V} exists {\displaystyle y\in B_{r-d(x,y)}(y)\subseteq B_{r}(x)} d d Balls is an open ball is the same thing ) standard topology on any normed vector space topology... B } is not an topology of metric spaces point result, the metric function might not be explicitly! 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