# examples of metric spaces with proofs

MOSFET blowing when soft starting a motor. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. $4)$Let (X,d) be a metric space.Prove that the collection of sets $T=\{A \subseteq X| \forall x \in A,\exists \epsilon>0$such that $B(x, \epsilon) \subseteq A\}$ is a topology on $X$.You need only to look the definition of a topolgy to solve this. 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 319.4 552.8 552.8 552.8 552.8 552.8 552.8 552.8 552.8 552.8 552.8 552.8 319.4 319.4 << /BaseFont/HWKPEX+CMMI12 It can be useful to isolate recurring pattern in our proofs that functions are metrics. /LastChar 196 $8)$A set $A$ in a metric (and topological in general)space is closed if $X$ \ $A$ is open. 0 0 0 0 0 0 691.7 958.3 894.4 805.6 766.7 900 830.6 894.4 830.6 894.4 0 0 830.6 670.8 $13)$Let $(X,d)$ be a metric space.Define $A+B=\{x+y|x \in A ,y \in B \}$ and $x+A=\{x+y| y \in A\}$ where $A,B \subseteq X$.Prove that if $A,B$ are open sets then $A+B,x+A$ are also open sets. 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 How late in the book-editing process can you change a characters name? Balls in sunﬂower metric d(x,y)= x −y x,y,0 colinear x+y otherwise centre (4,3), radius 6 MA222 – 2008/2009 – page 1.8 Subspaces, product spaces Subspaces. Do you need a valid visa to move out of the country? Is there a difference between a tie-breaker and a regular vote? To understand this concept, it is helpful to consider a few examples of what does and does not constitute a distance function for a metric space. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Then T is continuous if and only if T is bounded. << To learn more, see our tips on writing great answers. None. The closure of an open ball $B(a;\delta)$ is a subset of the closed ball centered at $a$ with radius $\delta$. Just for a bit of context, some of the proofs that I have done include: Can anybody give me any other (perhaps slightly more challenging) proofs to do about these topics? 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 Then f(x)∈ U and so there exists ε > 0 such that B(f(x),ε) ⊂ U. 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x,y) between every pair of points x,y ∈ X.The purpose of this chapter is to introduce metric spaces and give some deﬁnitions and examples. 761.6 272 489.6] /FontDescriptor 17 0 R endobj Consider the function dde ned above. rev 2020.12.10.38158, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. /LastChar 196 /LastChar 196 endobj i came up with some of these questions and the other questions where given by my proffesor to solve way back when i was attending a topology course.in conclusio these are some exercises i solved and i remembered and i choosed them for the O.P because they can be solved with the knowledge the O.P has learned so far (and mentions in his post).To help the O.P i also gave the appropriate definintions of some consepts used in the exercises. If they are from a book or other source, the source should be mentioned. 465 322.5 384 636.5 500 277.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The book is extremely rigorous and has hundreds of problems at varying difficulties; as with a lot of proofs, some take seconds, some might take you days. /Name/F6 1. My professor skipped me on christmas bonus payment. /FirstChar 33 525 768.9 627.2 896.7 743.3 766.7 678.3 766.7 729.4 562.2 715.6 743.3 743.3 998.9 How do I convert Arduino to an ATmega328P-based project? To show that f−1(U)is open, let x ∈ f−1(U). Have yoy learned about closures of sets in a metric space ,compactness ,sequences and completness? What important tools does a small tailoring outfit need? Suppose ﬁrst that T is bounded. << Let us go farther by making another deﬁnition: A metric space X is said to be sequentially compact if every sequence (xn)∞ 570 517 571.4 437.2 540.3 595.8 625.7 651.4 277.8] 319.4 575 319.4 319.4 559 638.9 511.1 638.9 527.1 351.4 575 638.9 319.4 351.4 606.9 /Name/F4 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 (c) Show that a continuous function from any metric space $Y$ to the space $X$ (with its discrete metric) must be constant. If $a\in X$ and $F$ is a closed subset of $X$ with $x\notin F$ then there exists $U, V$ open subsets of $X$ such that $x\in U,\ F\subseteq V$ and $U\cap V=\emptyset$. >> Uh...no. ... For appreciate the study of metric spaces in full generality, and for intuition, I request more useful examples of metric spaces that are significantly different from $\mathbb{R}^n$, and are not contain in $\mathbb{R}^n$. In a metric space $(X,d)$ with $x\in X,$ show that a sequence $(x_n)_{n\in \mathbb N}$ of members of $X$ satisfies $\lim_{n\to \infty}d(x,x_n)=0$ iff $\{n\in \mathbb N: d(x_n,x)\geq r\}$ is finite for every $r>0.$, (3.2). /Subtype/Type1 /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 /FontDescriptor 8 0 R Sequences and Convergence in Metric Spaces De nition: A sequence in a set X(a sequence of elements of X) is a function s: N !X. Show that (x, y ) ∈ R2 → (x + y , sin(x 2 y 3 )) ... with the same proof, in all metric spaces, the implication ⇐ is completely false in general metric spaces. ), (3.1). Assume that (x Solution: Xhas 23 = 8 elements. 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 611.8 816 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 680.6 777.8 736.1 555.6 722.2 750 750 1027.8 750 750 611.1 277.8 500 277.8 500 277.8 This distance function will satisfy a minimal set of axioms. Easily Produced Fluids Made Before The Industrial Revolution - Which Ones? Analysis on metric spaces 1.1. Definition. $3)$Let the space $C[0,1]=\{f[0,1] \rightarrow \mathbb{R}|f$ continuous on $[0,1]\}$ and $d(f,g)= \int_0^1|f(x)-g(x)|dx$. Theorem3.1–Productnorm Suppose X,Y are normed vector spaces. >> >> /LastChar 196 /FontDescriptor 11 0 R /FontDescriptor 23 0 R 863.9 786.1 863.9 862.5 638.9 800 884.7 869.4 1188.9 869.4 869.4 702.8 319.4 602.8 number of places where xand yhave di erent entries. 460 511.1 306.7 306.7 460 255.6 817.8 562.2 511.1 511.1 460 421.7 408.9 332.2 536.7 Metric spaces. >> /BaseFont/TKPGKI+CMBX10 /FontDescriptor 14 0 R d(x;y) is called the Hamming distance between xand y. There are several reasons: /BaseFont/AQLNGI+CMTI10 27 0 obj Suppose X,Y are normed vector spaces and let T :X → Y be linear. /BaseFont/VNVYCN+CMCSC10 /BaseFont/KCYEKS+CMBX12 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 277.8 777.8 472.2 472.2 777.8 Every metric space comes with a metric function. METRIC SPACES, TOPOLOGY, AND CONTINUITY Theorem 1.2. /FirstChar 33 If $X=\mathbb{R}$ and $d$ is the usual metric then every closed interval (or in fact any closed set) is the intersection of a family of open sets, i.e. 585.3 831.4 831.4 892.9 892.9 708.3 917.6 753.4 620.2 889.5 616.1 818.4 688.5 978.6 Proof. Because of this, the metric function might not be mentioned explicitly. >> /FirstChar 33 39 0 obj /Type/Font 743.3 743.3 613.3 306.7 514.4 306.7 511.1 306.7 306.7 511.1 460 460 511.1 460 306.7 /LastChar 196 We do not develop their theory in detail, and we leave the veriﬁcations and proofs as an exercise. If there is no source and you just came up with these, I think it would be appropriate to tell us much. /LastChar 196 /FirstChar 33 $A = \{f ∈ X | f(x) > 1,$ for $x \in [1/3, 2/3]\}$ is open in $X$. d(x n;x 1) " 8 n N . /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 The basic idea that we need to talk about convergence is to find a way of saying when two things are close. << /Subtype/Type1 First, suppose f is continuous and let U be open in Y. /LastChar 196 /Length 1963 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 It only takes a minute to sign up. /FirstChar 33 33 0 obj 638.9 638.9 958.3 958.3 319.4 351.4 575 575 575 575 575 869.4 511.1 597.2 830.6 894.4 /BaseFont/UAIIMR+CMR10 The inequality in (ii) is called the triangle inequality. >> /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 endobj G-metric topology coincides with the metric topology induced by the metric ‰G, which allows us to readily transform many concepts from metric spaces into the setting of G-metric space. Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions. $14)$Let $(X,d)$ be a metric space.A sequence $x_n \in X$ converges to $x$ if $\forall \epsilon >0 ,\exists n_0 \in \mathbb{N}$ such that $d(x_n,x)< \epsilon, \forall n \geqslant n_0$.Consider the space $(\mathbb{R}^m,d)$ with the euclideian metric.Prove that $x_n \rightarrow x=(x_1,x_2...x_m)$ in $\mathbb{R}^m$ if and only if $x_n^j \rightarrow x_j \in \mathbb{R}, \forall j \in \{1,2...m\}$(A sequence in $\mathbb{R}^m$ has the form $x_n=(x_n^1,x_n^2...x_n^m))$. A sequence (x n) in X is called a Cauchy sequence if for any ε > 0, there is an n ε ∈ N such that d(x m,x n) < ε for any m ≥ n ε, n ≥ n ε. Theorem 2. Any convergent sequence in a metric space is a Cauchy sequence. does not have to be defined at Example. /Name/F2 892.9 585.3 892.9 892.9 892.9 892.9 0 0 892.9 892.9 892.9 1138.9 585.3 585.3 892.9 575 1041.7 1169.4 894.4 319.4 575] << Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. 762.8 642 790.6 759.3 613.2 584.4 682.8 583.3 944.4 828.5 580.6 682.6 388.9 388.9 323.4 877 538.7 538.7 877 843.3 798.6 815.5 860.1 767.9 737.1 883.9 843.3 412.7 583.3 0 0 0 0 0 0 0 615.3 833.3 762.8 694.4 742.4 831.3 779.9 583.3 666.7 612.2 0 0 772.4 If $(X,d)$ is second countable, i.e. Let (X,d) be a metric space. A function f:X → Y between metric spaces is continuous if and only if f−1(U)is open in X for each set U which is open in Y. 593.8 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 For instance, the unique map from $\{0, 1\{\}$ with its usual topology to $\{0\}$ is constant, and continuous, but the domain is not connected. 15 0 obj /Widths[323.4 569.4 938.5 569.4 938.5 877 323.4 446.4 446.4 569.4 877 323.4 384.9 Example 2. 600.2 600.2 507.9 569.4 1138.9 569.4 569.4 569.4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0