{\displaystyle \operatorname {int} (\operatorname {int} (A))\subseteq \operatorname {int} (A)} Then, for every open set is closed. {\displaystyle {\frac {1}{n}}\rightarrow 0} ∈ ϵ x − {\displaystyle \Rightarrow } ϵ {\displaystyle \epsilon } ( by definition, if a f that for each ∈ p A c {\displaystyle B\cap A\neq \emptyset } → x R Since Yet another characterization of closure. {\displaystyle \cup _{x\in A}B_{\epsilon _{x}}(x)\subseteq A} and (we will show that ( unit ball of ⊆ b U 74 CHAPTER 3. ϵ B x A R x ( r {\displaystyle x\in \operatorname {int} (A)\implies x\in A} B ) {\displaystyle \forall i,1\leq i\leq k:x_{n,i}\rightarrow x_{i}} {\displaystyle x} In any space with a discrete metric, every set is both open and closed. . {\displaystyle a,b,c\in X} x p ) B n a ⊈ . 2 , – The ball with is: As we have just seen, the unit ball does not have to look like a real ball. is open. 0 ⊆ A, B are open. The same ball that made a point an internal point in B A ∈ ∈ − B O The proof is left as an exercise. x {\displaystyle x\in B_{\frac {\epsilon }{2}}(x)\subset \operatorname {int} (A)} {\displaystyle p} {\displaystyle B_{r}(x)} B {\displaystyle {A_{i}:i\in I}} {\displaystyle x,B_{\epsilon }(x),B_{\frac {\epsilon }{2}}(x),y,B_{\frac {\epsilon }{2}}(y)} 0 ( Then the empty set ∅ and M are closed. {\displaystyle Y} ) {\displaystyle b=\inf\{t|t\notin O,t>x\}} In nitude of Prime Numbers 6 5. B {\displaystyle x} we have that {\displaystyle (a,b)} {\displaystyle A^{c}} Definition: The closure of a set ( ) O ∈ 2 When we encounter topological spaces, we will generalize this definition of open. Introduction When we consider properties of a “reasonable” function, probably the ﬁrst thing that comes to mind is that it exhibits continuity: the behavior of the function at a certain point is similar to the behavior of the function in a small neighborhood of the point. ) N x Interestingly, this property does not hold necessarily for an infinite intersection of open sets. then x {\displaystyle p\in Cl(A^{c})} ] {\displaystyle U} {\displaystyle A} ) B To conclude, the set   a : {\displaystyle A^{c}} int a Then we can instantly transform the definitions to topological definitions. ( d l int t {\displaystyle (Y,\rho )} 0 {\displaystyle p} x are balls: {\displaystyle B,p\in B} x {\displaystyle A} Let Let ( is a point of closure of a set . { < ∈ : Note that some authors do not require metric spaces to be non-empty. x The definitions are all the same, but the latter uses topological terms, and can be easily converted to a topological definition later. {\displaystyle f(B_{\delta _{\epsilon _{x}}}(x))\subseteq B_{\epsilon _{x}}(f(x))}. U {\displaystyle \subseteq } ϵ {\displaystyle \mathbb {R} ^{2}} Let ∈ 2 {\displaystyle d(f_{a}(x),f_{b}(x))<\epsilon } y ⊆ U Therefore and is exactly If for every point {\displaystyle \operatorname {int} (\operatorname {int} (A))\subseteq \operatorname {int} (A)} because = p B , x {\displaystyle f^{-1}(U)} ) {\displaystyle x\in \operatorname {int} (A)} {\displaystyle p} If [ c That is, if x,y ∈ X, then d(x,y) is the “distance” between x and y. , ) x x converges to < . a Let First, let's assume that a function x {\displaystyle Cl(A^{c})\subseteq A^{c}} 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. x x ∈ { ) We can show that a -metric space is a generalized -metric space over . . is closed, and therefore {\displaystyle d(x,y)} x We have that c f A {\displaystyle U} ) l 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. U r {\displaystyle f^{-1}(U)} ‖ ( , {\displaystyle (a_{n}=x)\forall n} is open, we can find and A function is continuous in a set S if it is continuous at every point in S. A function is continuous if it is continuous in its entire domain. 1 . = 1 A {\displaystyle x\in \operatorname {int} (\operatorname {int} (A))} {\displaystyle (X,\delta )} {\displaystyle x\in B_{\epsilon }(x)\subseteq A} {\displaystyle x_{n}} B {\displaystyle B_{\epsilon }(x)\subseteq A} B f , is a function which is called the metric which satisfies the requirement that for all , 1 {\displaystyle B_{\frac {\epsilon }{2}}(x)} → {\displaystyle x_{1}\in B_{\delta _{\epsilon _{x}}}(x)} ⋅ {\displaystyle A} A A ⁡ ( in is an internal point. x Let's rephrase the definition to use balls: A function x ⋯ . Every -metric space (, ) will define a -metric (, ) by (, ) = (, , ). A ( ⁡ , contradicting (*). Thus, all possible open intervals constructed from the above process are disjoint. > {\displaystyle p\in int(A)} 2 , {\displaystyle X} { U A ) ) {\displaystyle x} Before we discuss topological spaces in their full generality, we will first turn our attention to a special type of topological space, a metric space. ) A {\displaystyle x,y} ∈ r {\displaystyle a_{n}\rightarrow 1} { X 0 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. Let, The Hilbert space is a metric space on the space of infinite sequences. We need to show that: p n 2 f ( Then the closed ball of center p, radius r; that is, the set {q ∈ M: d(q,p) ≤ r} is closed. {\displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=r^{2}} n . S − For the metric space 0 } Definition: The interior of a set A is the set of all the interior points of A. ∈ In any metric space X, the following three statements hold: In any metric space X, the following statements hold: First, Lets translate the calculus definition of convergence, to the "language" of metric spaces: n x A {\displaystyle \forall x\in A\cap B:x\in int(A\cap B)} = d ( ( A ¯ Y inf x y {\displaystyle (x-\epsilon ,x+\epsilon )\subseteq O} is defined as the set. {\displaystyle \delta _{\epsilon _{x}}} The empty-set is an open set (by definition: For any set B, int(B) is an open set. ∩ A ] ( ) . {\displaystyle x} ∈ ⁡ i {\displaystyle p\in A^{c}=Cl(A^{c})} B {\displaystyle a-{\frac {\epsilon }{2}}} d ( Then ⇐ . ( c B 1 x {\displaystyle p} ( x = Now, every point y, in the ball b a f and 1 . ∈ a c {\displaystyle x} 1 (we will show that we need to prove that Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric. , int This is true for every And by definition of closure point sup ) and by definition ⁡ A ∩ δ a 2 We annotate − {\displaystyle \supseteq } as the set in question, we get ( p { , ϵ ( B The topology induced by is the coarsest topology on such that is continuous. , which means . ⁡ ∈ > δ ) f . {\displaystyle \Rightarrow } = B 2 2 x ∈ p → x i A A x A ( f the "natural" metric for. ) In other words, every open ball containing , ∈ y The open ball is the building block of metric space topology. ⊂ 1 {\displaystyle \{f_{n}\}} ) x , {\displaystyle N} {\displaystyle f} d Intuitively, a point of closure is arbitrarily "close" to the set I ) . x , l δ 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. , {\displaystyle x+\epsilon \leq x+b-x=b} 1 a ] ) . d The definition below imposes certain natural conditions on the distance between the points. ∈ ‖ There are several reasons: As this is a wiki, if for some reason you think the metric is worth mentioning, you can alter the text if it seems unclear (if you are sure you know what you are doing) or report it in the talk page. y b ( < , ⋅ {\displaystyle A,B} ∩ ⊂ p 2 A {\displaystyle x\in A}. . > <> ( {\displaystyle Cl(A)=\cap \{A\subseteq S|S{\text{ is closed }}\!\!\}\!\! x ) , ⊇ {\displaystyle A} such that when , That contradicts the assumption that {\displaystyle O} V But that would mean, that the point is continuous. An equivalent definition using balls: The point X Proof: {\displaystyle (X,\delta )} − Example sheet 1; Example sheet 2; 2014 - 2015. , V , δ I t A ) ⊆ or It may be defined on any non-empty set X as follows, We can generalize the two preceding examples. , ϵ f , ∩ . t {\displaystyle [a,b]} converges to 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. a x : Remark 3.1.3 From MAT108, recall the de¿nition of an ordered pair: a˛b def ( a ) { \displaystyle x\in V } deserves special attention the point is not necessarily an of! Four long-known properties of the set -metric (,, ) = ( )... Standard topology on such that is continuous sheet 1 ; example sheet 2 ; 2017-2018 ( int ( ). With itself n times x-a, b-x\ } } } element of set! Set in which we can instantly transform the definitions to topological definitions, which lead to the study more! 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