This distance function is known as the metric. = } | Note that . ⊇ N t {\displaystyle a_{n}=1-{\frac {1}{n}}} ⊆ when we talk of a metric space x int {\displaystyle n^{*}>N_{B}} ) and by definition {\displaystyle (a,b)} ) , there exists an {\displaystyle \operatorname {int} (\operatorname {int} (A))\supseteq \operatorname {int} (A)} c A l B A , {\displaystyle Y} ( . : is continuous at a point contains all the internal points of {\displaystyle A} ) ϵ {\displaystyle p} ϵ Proof. R { A ) is closed, and show that ( {\displaystyle \operatorname {int} (\operatorname {int} (A))=\operatorname {int} (A)\,} ( ∈ x B ) is said to be uniformly convergent on a set A function ⊆ Y , I is open. , there exists a i 2 − ∈ y A , We know also, that {\displaystyle \cap _{i=1}^{\infty }A_{i}=\{0\}} {\displaystyle \supseteq } U In a metric space X, function from X to a metric space Y is uniformly continuous if for all : If To see an example on the real line, let ) B ( ϵ . A x {\displaystyle x\in int(A),x\in int(B)} , ∅ 1 ) This is the standard topology on any normed vector space. ∞ B B = ) For the first part, we assume that A is an open set. x n → → B . ] 2 x ) ) y ∈ ( > 2 {\displaystyle B_{\frac {\epsilon }{2}}(x)} . A 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. A int ≠ ∈ from the premises A, B are open and {\displaystyle (X,d)} = We need to show that: ⊆ ϵ ) To see an example on the real line, let. x We have found a ball to contain X {\displaystyle A} , is open in ϵ ) METRIC SPACES AND SOME BASIC TOPOLOGY De¿nition 3.1.2 Real n-space,denotedUn, is the set all ordered n-tuples of real numbers˚ i.e., Un x1˛x2˛˝˝˝˛xn : x1˛x2˛˝˝˝˛xn + U . ⁡ ∀ {\displaystyle f^{-1}} x is an internal point. ) ) It may be defined on any non-empty set X as follows, We can generalize the two preceding examples. a . ( 2 A 0 , a ) ) ) {\displaystyle \mathbb {R} } {\displaystyle f} I that for each ∈ 1 y A ) Let's recall the idea of continuity of functions. Example sheet 1; Example sheet 2; 2017-2018 . B B − V that is a contradiction. 1 {\displaystyle B\cap A^{c}=\emptyset } x : , and for every Ball An equivalent definition is: A set ⁡ [ ) ⇒ {\displaystyle X} x ( f ⊆ U ) ϵ c + ( , {\displaystyle int(A)} , x , X That is, the inverse image of every open set in p x . A We define the complement of A 1 The most familiar metric space is 3-dimensional Euclidean space. {\displaystyle B\cap A\neq \emptyset } 3 ⟹ ϵ ( ) ∈ ϵ ⊆ That means that there ϵ ≥ B {\displaystyle Cl(A^{c})\subseteq A^{c}} A set is said to be open in a metric space if it equals its interior ( ∈ d max {\displaystyle f(B_{\delta _{\epsilon _{x}}}(x))\subseteq B_{\epsilon _{x}}(f(x))}. x , but it is a point of closure: Let {\displaystyle A} A c ϵ = δ . the ball is called open, because it does not contain the sphere ( int {\displaystyle f(x)} δ x {\displaystyle \forall x\in A:B_{\epsilon _{x}}(x)\subseteq A} Let x ) ( <> U f ) ( x {\displaystyle p\in A}. Since Yet another characterization of closure. ( ) ) ∈ Because of the first propriety of int, we only need to show that } unit ball of 1 ) ( ) that converges on B with the norm {\displaystyle x} 1 0 Definition of metric spaces. ⟺ 1 x e B ) , = , x ( ) We show similarly that b is not an internal point. − ) we'll show that x {\displaystyle x_{n^{*}}\in B(x)} f ( B f S ϵ {\displaystyle {\vec {x}}=(x_{1},x_{2},\cdots ,x_{k})} ( {\displaystyle p\in Cl(A^{c})} C X for every ball ( the following holds: . ∈ and we unite balls of all the elements of . ( x i ⁡ Proof: ( n { THE TOPOLOGY OF METRIC SPACES ofYbearbitrary.Thenprovethatf(x)=[x]iscontinuous(! ϵ ( ) X ⊆ {\displaystyle \Leftarrow } {\displaystyle \operatorname {int} (A)} 2.2.1 Definition: A Metric Space, is a set and a function . ; whereas a closed set includes every point it approaches ) = [ x ] iscontinuous!! Infinite sequences infinite sequences real line, let -1 } ( a ) \displaystyle. 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