is it possible to read and play a piece that's written in Gflat (6 flats) by substituting those for one sharp, thus in key G? During many proofs, I visualize something like $\mathbb{R}^2$. >> /BaseFont/AZRCNF+CMMI10 >> /BaseFont/JKPQDT+CMSY7 Cauchy Sequences and Complete Metric Spaces Let’s rst consider two examples of convergent sequences in R: Example 1: Let x n = 1 n p 2 for each n2N. Uh...no. d(x n;x 1) " 8 n N . 575 575 575 575 575 575 575 575 575 575 575 319.4 319.4 350 894.4 543.1 543.1 894.4 But I'm getting there! 277.8 500] 323.4 354.2 600.2 323.4 938.5 631 569.4 631 600.2 446.4 452.6 446.4 631 600.2 815.5 /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 To show that f−1(U)is open, let x ∈ f−1(U). Proof. Functional Analysis by Prof. P.D. (2). Example 1.1.2. Assume that (x 646.5 782.1 871.7 791.7 1342.7 935.6 905.8 809.2 935.9 981 702.2 647.8 717.8 719.9 /Subtype/Type1 number of places where xand yhave di erent entries. If $f:(X,d)\to (X,d)$ is continuous and $f\circ f=f$ then $f(X)$ is closed. (0, 1) is a closed and bounded subset of the space (0, 1). /FontDescriptor 38 0 R A metric space is an ordered pair (,) where is a set and is a metric on , i.e., a function: × → such that for any ,, ∈, the following holds: 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 /Widths[350 602.8 958.3 575 958.3 894.4 319.4 447.2 447.2 575 894.4 319.4 383.3 319.4 /Type/Font We usually denote s(n) by s n, called the n-th term of s, and write fs ngfor the sequence, or fs 1;s 2;:::g. See the nice introductory paragraphs about … 1. /Filter[/FlateDecode] (2.2). Where are these questions from? endobj How to prove this statement for metric subspaces? /BaseFont/ZCGRXQ+CMR8 stream Definition. /FirstChar 33 $4)$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. << 761.6 272 489.6] Example. Srivastava, Department of Mathematics, IIT Kharagpur. The book is logically organized and the exposition is clear. endobj /FontDescriptor 35 0 R My professor skipped me on christmas bonus payment. $11)$Let $(X,d)$ be a metric space .We define the diameter of a set $A$ as $diam(A)=\sup \{d(x,y)|x,y \in A\}$.Suppose that $B$ is a bounded subset of X and $C \subseteq B$.Prove that $diam(C) \leqslant diam(B)$. >> /FontDescriptor 23 0 R Show that $$f(u,v)=d(u,v)+e(f(u),f(v)) \quad \text {for } u,v\in X$$ is a metric on $X$ equivalent to $d.$ (In particular, with $Y=\mathbb R$ and $e(y,y')=|y-y'|,$ this is useful in constructions for other problems and examples. $12)$Let $X$ be the space of continuous functions on $[0, 1]$($C[0,1]$) with the metric $d(f,g)= \sup_{x \in [0,1]}|f(x)-g(x)|$.Show that $d$ is indeed a metric. /FontDescriptor 29 0 R Have yoy learned about closures of sets in a metric space ,compactness ,sequences and completness? /Name/F6 Note that each x n is an irrational number (i.e., x n 2Qc) and that fx ngconverges to 0.Thus, fx ngconverges in R (i.e., to an element of R).But 0 is a rational number (thus, 0 62Qc), so although the sequence fx /Subtype/Type1 Let ( M;d ) be a metric space and ( x n)n 2 N 2 M N.Then we de ne (i) x n! Show that if $F$ is a family of subsets of a metric space such that $\cup G$ is closed whenever $G$ is a countable subset of $F$ , then $\cup F$ is closed. 813.9 813.9 669.4 319.4 552.8 319.4 552.8 319.4 319.4 613.3 580 591.1 624.4 557.8 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 The inequality in (ii) is called the triangle inequality. /Type/Font 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. 30 0 obj $A = \{f ∈ X | f(x) > 1,$ for $x \in [1/3, 2/3]\}$ is open in $X$. Prove that the set $\mathbb{Z}$ is a closed subsets of the real line under the usual metric.Also prove that the set of rational numbers in not closed under the same metric. /Subtype/Type1 There is nothing original in this problems list. /Widths[1138.9 585.3 585.3 1138.9 1138.9 1138.9 892.9 1138.9 1138.9 708.3 708.3 1138.9 Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. 877 0 0 815.5 677.6 646.8 646.8 970.2 970.2 323.4 354.2 569.4 569.4 569.4 569.4 569.4 /Subtype/Type1 843.3 507.9 569.4 815.5 877 569.4 1013.9 1136.9 877 323.4 569.4] /BaseFont/TKPGKI+CMBX10 /Subtype/Type1 /Subtype/Type1 27 0 obj << (c) Show that a continuous function from any metric space $Y$ to the space $X$ (with its discrete metric) must be constant. /Name/F8 << 343.8 593.8 312.5 937.5 625 562.5 625 593.8 459.5 443.8 437.5 625 593.8 812.5 593.8 Proposition 2.1 A metric space X is compact if and only if every collection F of closed sets in X with the ﬁnite intersection property has a nonempty intersection. /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 638.9 638.9 958.3 958.3 319.4 351.4 575 575 575 575 575 869.4 511.1 597.2 830.6 894.4 /FirstChar 33 A lot of really good metric problems have already been posted, but I'd like to add that you may want to try Topology Without Tears by Sidney A. Morris. endobj 12 0 obj 424.4 552.8 552.8 552.8 552.8 552.8 813.9 494.4 915.6 735.6 824.4 635.6 975 1091.7 388.9 1000 1000 416.7 528.6 429.2 432.8 520.5 465.6 489.6 477 576.2 344.5 411.8 520.6 /FirstChar 33 Prove or disprove two statements about open functions on metric spaces, Proving the Hausdorff property for $\kappa$-metric spaces, metric spaces proving the boundary of A is closed, Metric Space defined by an Infinite Sequence of Metric Spaces in this case not a Metric Space. (1.1). /LastChar 196 %PDF-1.2 Note . $5)$ Prove that the set of rational numbers is not an open subset of $\mathbb{R}$ under the metric $d(x,y)=|x-y|$(usual metric), $6)$Prove that the set $A=\{(x,y) \in \mathbb{R}^2|x+y>1\}$ is an open set in $\mathbb{R}^2$ under the metric $d((x_1,y_1),(x_2,y_2))=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$. >> Let be a mapping from to We say that is a limit of at , if 0< . 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 How is this octave jump achieved on electric guitar? MathJax reference. /Type/Font (b) Show that every function from $X$ with its discrete metric to any metric space $Y$ is in fact continuous. Deﬁnition 2 (absolute value function). 306.7 766.7 511.1 511.1 766.7 743.3 703.9 715.6 755 678.3 652.8 773.6 743.3 385.6 There is nothing original in this problems list. Let (X,d) be a metric space. Proof. /BaseFont/KCYEKS+CMBX12 1135.1 818.9 764.4 823.1 769.8 769.8 769.8 769.8 769.8 708.3 708.3 523.8 523.8 523.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 627.2 817.8 766.7 692.2 664.4 743.3 715.6 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 I am going to explore open sets, metric spaces, and also closed sets. For example, if = = Stanisław Ulam, then (,) =. /Name/F7 9 0 obj 460 664.4 463.9 485.6 408.9 511.1 1022.2 511.1 511.1 511.1 0 0 0 0 0 0 0 0 0 0 0 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 541.2 833 666.2 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. 874 706.4 1027.8 843.3 877 767.9 877 829.4 631 815.5 843.3 843.3 1150.8 843.3 843.3 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] /LastChar 196 Replace each metric with the derived bounded metric. /Type/Font If they are from a book or other source, the source should be mentioned. Roughly, the "metric spaces" we are going to study in this module are sets on which a distance is defined on pairs of points. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. /Name/F3 To subscribe to this RSS feed, copy and paste this URL into your RSS reader. /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 298.4 878 600.2 484.7 503.1 446.4 451.2 468.8 361.1 572.5 484.7 715.9 571.5 490.3 Some important properties of this idea are abstracted into: Definition A metric space is a set X together with a function d (called a metric or "distance function") which assigns a real number d(x, y) to every pair x, y X satisfying the properties (or axioms): 826.4 295.1 531.3] /Subtype/Type1 /LastChar 196 Then if we de ne the distance of two points in distinct spaces of the disjoint union to be 1, then the result is a metric space. Solution: Xhas 23 = 8 elements. /BaseFont/HWKPEX+CMMI12 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 0 683.3 902.8 844.4 755.5 A metric space consists of a set Xtogether with a function d: X If there is no source and you just came up with these, I think it would be appropriate to tell us much. /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 >> Let $d,e$ be metrics on $X$ such that there exist positive $k,k'$ such that $d(u,v)\leq k\cdot e(u,v)$ and $e(u,v)\leq k'\cdot d(u,v)$ for all $u,v \in X.$ Show that $d,e$ are equivalent. 1138.9 1138.9 892.9 329.4 1138.9 769.8 769.8 1015.9 1015.9 0 0 646.8 646.8 769.8 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 << Proof. The "discrete metric" on a space $X$ is one in which $d(x, y) = 1$ if $x \ne y$, and $d(x, x) = 0$. /FirstChar 33 Can I combine two 12-2 cables to serve a NEMA 10-30 socket for dryer? /BaseFont/AQLNGI+CMTI10 COMPACT SETS IN METRIC SPACES NOTES FOR MATH 703 ANTON R. SCHEP In this note we shall present a proof that in a metric space (X;d) a subset Ais compact if and only if it is sequentially compact, i.e., if every sequence in Ahas a convergent subsequence with limit in A. None. /LastChar 196 Co-requisites. 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 Metric spaces (definition, examples, open sets, closed sets, interior, closure, limit points, ... MATH10011 Analysis and MATH10010 Introduction to Proofs and Group Theory . 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 /Subtype/Type1 Many problems in pure and applied mathematics reduce to a problem of common fixed point of some self-mapping operators which are defined on metric spaces. 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 39 0 obj /Name/F10 If $(X,d)$ is second countable, i.e. @RamizKaraeski No, not yet. for each $x\in X,$ there exists a countable family $\eta(x)$ of open sets such that for any open neighborhood $U$ of $x$, there exists $V\in \eta(x)$ such that $x\in U\subseteq V$. In particular we will be able to apply them to sequences of functions. 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. /LastChar 196 Proofs covered in class P. Karageorgis pete@maths.tcd.ie 1/22. Definition and examples of metric spaces. 21 0 obj 523.8 585.3 585.3 462.3 462.3 339.3 585.3 585.3 708.3 585.3 339.3 938.5 859.1 954.4 $13)$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. Every metric space comes with a metric function. /FontDescriptor 20 0 R 535.6 641.1 613.3 302.2 424.4 635.6 513.3 746.7 613.3 635.6 557.8 635.6 602.2 457.8 We do not develop their theory in detail, and we leave the veriﬁcations and proofs as an exercise. $8)$A set $A$ in a metric (and topological in general)space is closed if $X$ \ $A$ is open. /Subtype/Type1 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 One of the generalizations of metric spaces is the partial metric space in which self-distance of points need not to be zero but the property of symmetric and modified version of triangle inequality is satisfied. Different metrics that generate the same topology are called equivalent metrics: (2.1). Complete Metric Spaces Deﬁnition 1. First, suppose f is continuous and let U be open in Y. 727.8 813.9 786.1 844.4 786.1 844.4 0 0 786.1 552.8 552.8 319.4 319.4 523.6 302.2 /Type/Font This distance function will satisfy a minimal set of axioms. << /LastChar 196 Is a countable intersection of open sets always open? 766.7 715.6 766.7 0 0 715.6 613.3 562.2 587.8 881.7 894.4 306.7 332.2 511.1 511.1 MOSFET blowing when soft starting a motor. In fact, later we will see that if f„ ;” is continuous, then lim f„xn;yn” f„x;y”.The previous two theorems are examples of this with f„x;y” x + y and f„c;x” cx, Let us go farther by making another deﬁnition: A metric space X is said to be sequentially compact if every sequence (xn)∞ 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 319.4 958.3 638.9 575 638.9 606.9 473.6 453.6 447.2 638.9 606.9 830.6 606.9 606.9 Easily Produced Fluids Made Before The Industrial Revolution - Which Ones? 41 0 obj Please check again that all these are "standard results". And give an example of two equivalent metrics that are not uniformly equivalent. .It would be helpfull for the O.P to be introduced and to work with new consepts in these exercises and in exercises in general. endobj 33 0 obj (If such $k,k'$ exist then $d,e$ are called uniformly equivalent). /BaseFont/QLOALX+CMR7 Let $(X,d)$ and $(Y,e)$ be metric spaces and let $f:X\to Y$ be continuous. /Widths[295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 413.2 531.3 826.4 295.1 354.2 Let . The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. ), (3.1). 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 /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 706.4 938.5 877 781.8 754 843.3 815.5 877 815.5 endobj For many purposes, the example of R2 with the usual distance function is precisely the one you should have in mind when thinking about metric spaces in general. >> /FirstChar 33 /BaseFont/CFYOEN+CMR12 Let $f:(X,d)\to (Y,d')$, $a\in X$, and let $\beta_{f(a)}$ be a basis for the neighborhood system at $f(a)$. /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 In 1906 Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel. /BaseFont/UAIIMR+CMR10 Well most of the questions posed here are rather "theorems" that I was given (to prove as exercises) when I was learning topology at university and I just typed them here by memory. 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. 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 Remark. If $X=\mathbb{R}$ and $d$ is the usual metric then every open subset of $X$ is at most a countable union of disjoint open intervals. 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. 570 517 571.4 437.2 540.3 595.8 625.7 651.4 277.8] 869.4 818.1 830.6 881.9 755.6 723.6 904.2 900 436.1 594.4 901.4 691.7 1091.7 900 /FontDescriptor 17 0 R a $G_\delta$ set. Prove that $f$ is continuous at $a$ iff $f^{-1}(N)$ is a neighborhood of $a$ for each $N \in \beta_{f(a)}$. if there exists a countable family $\mathcal{B}$ of open sets in $(X,d)$ such that for each open set $U$ in $X$, there exists an open set $V\in \mathcal{B}$ such that $V\subseteq U$, then $(X,d)$ is first countable but the converse is not necessarily true. 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. The most familiar is the real numbers with the usual absolute value. 591.1 613.3 613.3 835.6 613.3 613.3 502.2 552.8 1105.5 552.8 552.8 552.8 0 0 0 0 By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. /LastChar 196 One measures distance on the line R by: The distance from a to b is |a - b|.. Left as an exercise. a metric will be called the triangle inequality since in the case of R2 it says exactly that the length of one side of a triangle is less than or equal to the sum of the lengths of the other two sides. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 600.2 600.2 507.9 569.4 1138.9 569.4 569.4 569.4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /FirstChar 33 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 Also show that the subset What important tools does a small tailoring outfit need? /Name/F2 Example 1.1.3. 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 /Subtype/Type1 :D. General advice. /FirstChar 33 d(x;y) is called the Hamming distance between xand y. 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. Is there a difference between a tie-breaker and a regular vote? I've just finished learning about metric spaces, continuity, and open balls about points in metric spaces. << For the theory to work, we need the function d to have properties … 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 /LastChar 196 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. << >> endobj Metric spaces. I am going to move on to the concept of Coarse Geometry and Topology together with their applications. $9)$A subset $Y$ of metric space X is connected if there DO NOT exist two open sets $A,B \subseteq X$ such that $Y=A \cup B$ and $A \cap B= \emptyset$. 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 295.1 endobj 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 METRIC SPACES, TOPOLOGY, AND CONTINUITY Theorem 1.2. /FontDescriptor 14 0 R /FirstChar 33 Definition. This book provides a wonderful introduction to metric spaces, highly suitable for self-study. Do you need a valid visa to move out of the country? Then T is continuous if and only if T is bounded. >> The closure of an open ball $B(a;\delta)$ is a subset of the closed ball centered at $a$ with radius $\delta$. It only takes a minute to sign up. 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 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? A sequence hxni1 n=1 in a G-metric space (X;G) is said to be G-convergent with limit p 2 X if it converges to p in the G-metric topology, ¿(G). 306.7 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 306.7 306.7 One motivation for doing this is to extend definitions and results from the analysis of functions of a single real variable (the topic of the Convergence and Continuity module) to a more general setting. 692.5 323.4 569.4 323.4 569.4 323.4 323.4 569.4 631 507.9 631 507.9 354.2 569.4 631 For a metric $d,$ show that $e_1=d/(1+d)$ and $e_2=\min (1,d)$ are metrics and are equivalent to $d.$, (2.3). If $(X,d)$ is a metric space and $a\in X$, for each $\delta \gt 0$, the open ball $B(a; \delta)$ is a neighborhood of each of its points. 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). /FirstChar 33 /Widths[306.7 514.4 817.8 769.1 817.8 766.7 306.7 408.9 408.9 511.1 766.7 306.7 357.8 Throughout this chapter we will be referring to metric spaces. >> A metric space is an ordered pair (X;ˆ) such that X is a set and ˆ is a metric on X. 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 Characterization of the limit in terms of sequences. $1)$Prove or disprove with a counterxample: Making statements based on opinion; back them up with references or personal experience. $7)$Let $(X,d)$ be a metric space and $A \subset X$.We define $(x_0,A)=\inf\{d(x_0,y)|y \in A \}$. How late in the book-editing process can you change a characters name? /Name/F4 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 endobj $\endgroup$ – Janitha357 Jul 16 '17 at 16:32 493.6 769.8 769.8 892.9 892.9 523.8 523.8 523.8 708.3 892.9 892.9 892.9 892.9 0 0 36 0 obj METRIC SPACES 1.1 Deﬁnitions and examples As already mentioned, a metric space is just a set X equipped with a function d : X×X → R which measures the distance d(x,y) beween points x,y ∈ X. /FontDescriptor 8 0 R 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 $U\cap V=\emptyset$. 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 The function d is called the metric on X.It is also sometimes called a distance function or simply a distance.. Often d is omitted and one just writes X for a metric space if it is clear from the context what metric is being used.. We already know a few examples of metric spaces. We say ˆ is a metric on X if ˆ: X X ! Because of this, the metric function might not be mentioned explicitly. /Type/Font Let X be a set. << 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 For any space X, let d(x,y) = 0 if x = y and d(x,y) = 1 otherwise. 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 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. 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. >> To understand what exactly coarse geometry and topology are, there are a number of definitions that I need to explore. For example, I think the first question is a special case of "Retract of a Hausdorff space is closed", and the ones before the last are about the normality and regularity of metric spaces. Every point of $X$ has a countable neighborhood base, i.e. x 1 (n ! $3)$Let the space $C[0,1]=\{f[0,1] \rightarrow \mathbb{R}|f$ continuous on $[0,1]\}$ and $d(f,g)= \int_0^1|f(x)-g(x)|dx$. 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 Then f(x)∈ U and so there exists ε > 0 such that B(f(x),ε) ⊂ U. 844.4 319.4 552.8] However the name is due to Felix Hausdorff.. Metric Spaces Worksheet 1 ... Now we are ready to look at some familiar-ish examples of metric spaces. Metric spaces: definition and examples. 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 /Widths[319.4 552.8 902.8 552.8 902.8 844.4 319.4 436.1 436.1 552.8 844.4 319.4 377.8 Deﬂnition 1.7. 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 Whenever you prove a theorem, try to see what happens if you weaken the hypotheses - look for a counterexample or try to get away with assuming less. 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 For instance, the unique map from $\{0, 1\{\}$ with its usual topology to $\{0\}$ is constant, and continuous, but the domain is not connected. 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 << $15)$Let a function $f:(X,d_1) \rightarrow (Y,d_2)$.Prove that $f$ is continuous in $X$ if and only if for every sequence $x_n \rightarrow x$ in $X$ we have $f(x_n) \rightarrow f(x)$ in $Y$. /LastChar 196 Every sequence in $(X,d)$ converges to at most one point in $X$. 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 0 663.6 885.4 826.4 736.8 << /Type/Font << Show that if $\lim_{n\to \infty} d(x,x_n)=0=\lim_{n\to \infty}d(x,x'_n)$ then $\lim_{n\to \infty}d(x_n,x'_n)=0.$, (3.3). 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 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 This metric, called the discrete metric, satisﬁes the conditions one through four. 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