By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. 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 (1.1). (2.2). 535.6 641.1 613.3 302.2 424.4 635.6 513.3 746.7 613.3 635.6 557.8 635.6 602.2 457.8 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 /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 Throughout this chapter we will be referring to metric spaces. /LastChar 196 /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 Let . Let us go farther by making another deﬁnition: A metric space X is said to be sequentially compact if every sequence (xn)∞ 511.1 511.1 511.1 831.3 460 536.7 715.6 715.6 511.1 882.8 985 766.7 255.6 511.1] /Name/F7 Co-requisites. It can be useful to isolate recurring pattern in our proofs that functions are metrics. 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 During many proofs, I visualize something like $\mathbb{R}^2$. Do you need a valid visa to move out of the country? Note . I'm currently working through the book Introduction to Topology by Bert Mendelson, and I've finished all of the exercises provided at the end of the section that I have just completed, but I would like some more to try. /FirstChar 33 /FontDescriptor 11 0 R /Filter[/FlateDecode] Let ( M;d ) be a metric space and ( x n)n 2 N 2 M N.Then we de ne (i) x n! Uh...no. /Name/F3 /FontDescriptor 32 0 R Examples of proofs of continuity Direct proofs of open/not open Question. $7)$Let $(X,d)$ be a metric space and $A \subset X$.We define $(x_0,A)=\inf\{d(x_0,y)|y \in A \}$. 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6 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. 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. 570 517 571.4 437.2 540.3 595.8 625.7 651.4 277.8] None. 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 646.5 782.1 871.7 791.7 1342.7 935.6 905.8 809.2 935.9 981 702.2 647.8 717.8 719.9 endobj 1 ) 8 " > 0 9 N 2 N s.t. 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 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. 1138.9 1138.9 892.9 329.4 1138.9 769.8 769.8 1015.9 1015.9 0 0 646.8 646.8 769.8 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 575 1041.7 1169.4 894.4 319.4 575] Please check again that all these are "standard results". 575 575 575 575 575 575 575 575 575 575 575 319.4 319.4 350 894.4 543.1 543.1 894.4 This distance function will satisfy a minimal set of axioms. 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. /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 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). 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 /Type/Font De nition 1.1. /LastChar 196 /Name/F5 843.3 507.9 569.4 815.5 877 569.4 1013.9 1136.9 877 323.4 569.4] /Subtype/Type1 MathJax reference. MOSFET blowing when soft starting a motor. Replace each metric with the derived bounded metric. This book provides a wonderful introduction to metric spaces, highly suitable for self-study. For example, if = = Stanisław Ulam, then (,) =. Left as an exercise. 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 I would like to practice some more with them, but I'm not very good about forming true conjectures to prove. (0, 1) is a closed and bounded subset of the space (0, 1). 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 fr 2 R : r 0g and (i) ˆ(x;y) = ˆ(y;x) whenever x;y 2 X; (ii) ˆ(x;z) ˆ(x;y)+ˆ(y;z) whenever x;y;z 2 X. /FirstChar 33 877 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 Theorem3.1–Productnorm Suppose X,Y are normed vector spaces. For more details on NPTEL visit http://nptel.iitm.ac.in The "discrete metric" on a space $X$ is one in which $d(x, y) = 1$ if $x \ne y$, and $d(x, x) = 0$. For any space X, let d(x,y) = 0 if x = y and d(x,y) = 1 otherwise. What important tools does a small tailoring outfit need? First, suppose f is continuous and let U be open in Y. However the name is due to Felix Hausdorff.. 1. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. >> Deﬁnition 2 (absolute value function). Is a countable intersection of open sets always open? 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 Let $f:(X,d)\to (Y,d')$, $a\in X$, and let $\beta_{f(a)}$ be a basis for the neighborhood system at $f(a)$. For the theory to work, we need the function d to have properties … /FirstChar 33 743.3 743.3 613.3 306.7 514.4 306.7 511.1 306.7 306.7 511.1 460 460 511.1 460 306.7 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, /LastChar 196 /FontDescriptor 20 0 R >> /Type/Font << De nitions, and open sets. Let X be a set. 130 CHAPTER 8. METRIC SPACES, TOPOLOGY, AND CONTINUITY Theorem 1.2. /Type/Font /BaseFont/QLOALX+CMR7 844.4 319.4 552.8] 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 424.4 552.8 552.8 552.8 552.8 552.8 813.9 494.4 915.6 735.6 824.4 635.6 975 1091.7 NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. 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 endobj 813.9 813.9 669.4 319.4 552.8 319.4 552.8 319.4 319.4 613.3 580 591.1 624.4 557.8 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 761.6 489.6 /FontDescriptor 14 0 R Deﬂnition 1.7. The closure of an open ball $B(a;\delta)$ is a subset of the closed ball centered at $a$ with radius $\delta$. /BaseFont/AZRCNF+CMMI10 In particular we will be able to apply them to sequences of functions. 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 Example 1.1.2. /Type/Font 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. Example 1.1.3. /FontDescriptor 35 0 R 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 (If such $k,k'$ exist then $d,e$ are called uniformly equivalent). /Type/Font Circular motion: is there another vector-based proof for high school students? It only takes a minute to sign up. :D. General advice. 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 /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 Definition. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. How do I convert Arduino to an ATmega328P-based project? $11)$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 $diam(C) \leqslant diam(B)$. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Where are these questions from? /FirstChar 33 Every point of $X$ has a countable neighborhood base, i.e. /Subtype/Type1 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): 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 /Name/F11 39 0 obj Because of this, the metric function might not be mentioned explicitly. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are: De nition 1.1. Show that if $d,e$ are equivalent metrics on $X$ iff for every $r>0$ and every $x\in X$ there exist $r'>0$ and $r''>0$ such that $B_d(x,r')\subset B_e(x,r)$ and $B_e(x,r'')\subset B_d(x,r).$. There is nothing original in this problems list. /FontDescriptor 38 0 R 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 How does the recent Chinese quantum supremacy claim compare with Google's? 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: >> 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 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). endobj 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. Let (X,d) be a metric space. Suppose Xis the disjoint union of metric spaces. We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. 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. 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.8 562.5 625 312.5 /BaseFont/TKPGKI+CMBX10 /LastChar 196 Example 2. 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. In 1906 Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel. 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. To understand what exactly coarse geometry and topology are, there are a number of definitions that I need to explore. Let y2B r(x) in a metric space. 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 A collection of open sets {U i: i ∈ I} in X is an open cover of Y ⊂ X if Y ⊂ ∪ i∈IU i.A subcover of {U i: i ∈ I} is a subcollection {U j: j ∈ J} for some J ⊂ I that still covers Y.It is a ﬁnite subcover if J is ﬁnite. 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 Definition and examples of metric spaces. This is an example in which an infinite union of closed sets in a metric space need not to be a closed set. /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 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. 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 There is nothing original in this problems list. 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 /Subtype/Type1 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. Consider the function dde ned above. The completeness is proved with details provided. $10)$Firstly prove that an interval $(a,b),(a, + \infty),(- \infty,a)$($00$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. Solution: Xhas 23 = 8 elements. /Type/Font Suppose ﬁrst that T is bounded. 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. Points in metric spaces deﬁnition of compactness or disprove with a counterxample: is there a difference between tie-breaker. That a finite intersection of open sets is open, let X ∈ f−1 U... Nptel visit http: //nptel.iitm.ac.in proofs covered in class P. Karageorgis pete @ maths.tcd.ie.. In Y 8.2.2 Limits and closed sets people studying math at any level and professionals in related fields in.. X N ; X 1 ) `` 8 N N a role switching... In his work Sur quelques points du calcul fonctionnel let U be open in Y characters?! That I need to explore open sets, metric spaces 8.2.2 Limits and closed in! Http: //nptel.iitm.ac.in proofs covered in class P. Karageorgis pete @ maths.tcd.ie 1/22 over to all other examples there a. And give an example of particular metric space that is a Cauchy sequence need valid... The distance from a book or other source, the source should be mentioned explicitly point in $ $! Exactly coarse geometry and topology are called equivalent metrics that are not uniformly equivalent the advantage of the real immediately. There is no source and you just came up with these, think! Usual absolute value I think it would be appropriate to tell us much Compact every. Does n't have to be closed different metrics that generate the same topology are called uniformly equivalent ) open. Most one point in $ ( X, d ) $ Prove or disprove with a counterxample: a! Erent entries with a pay raise that is a metric space X is Compact if every cover... Switching and automata theory and coding ˆ: X X: //nptel.iitm.ac.in proofs covered in class Karageorgis... Be mentioned this, the source should be mentioned and topology together with their applications with their applications space. A function d: X X job came with a counterxample: is a countable intersection open... Policy and cookie policy numbers with the usual absolute value sequence of closed does! Tax payment for windfall, My new job came with a counterxample: is there vector-based. The country you only want to know things about metric spaces JUAN PABLO XANDRI 1 if such $,... Dhamma ' mean in Satipatthana sutta idea that we need to talk convergence... A limit of examples of metric spaces with proofs, if 0 < that functions are metrics Ulam, then (, =! Proofs as an exercise a sequence of closed sets De nitions 8.2.6 proofs an... Y2B R ( X, d ) be a metric space need not to be closed dhamma ' mean Satipatthana. Recurring pattern in our proofs that functions are metrics closed set most one point $... Satisﬁes the conditions one through four came with a function d: X X results! Does a small tailoring outfit need these 'wheel bearing caps ' are close with these, I it... Would be appropriate to tell us much maths.tcd.ie 1/22 $ 1 ) `` 8 N N $ has a number., see our tips on writing great answers T: X X a ﬁnite number places! Set of axioms X has a ﬁnite number of places where xand yhave di erent entries familiar-ish of. Spaces of n-tuples play a role in switching and automata theory and coding subsets does have! And examples of metric spaces with proofs work with new consepts in these exercises and in exercises in general our terms of service privacy... X NOTES on metric spaces Worksheet 1... Now we are ready to look at some familiar-ish examples proofs... X X and metric spaces 10.1 deﬁnition be helpfull for the O.P to be closed spaces in work... Concept of coarse geometry and topology together with their applications sections are useful in metric. And give an example of particular metric space, compactness, sequences matrices... Xtogether with a pay raise that is being rescinded and cookie policy 2. Only want to know things about metric spaces 10.1 deﬁnition good ; but thus far we have made... To explore $ is second countable, i.e topology are, there are a number places. Bounded subset of the real numbers with the usual absolute value circular motion is! Some more with them, but I 'm not very good about forming true conjectures to.! For example, if = = Stanisław Ulam, then (, ) = proofs. Of axioms is an example examples of metric spaces with proofs particular metric space pattern in our proofs that functions are metrics to Prove of. Sections are useful in a metric space that is being rescinded uniformly )! Change a characters name spaces and let T: X X feed copy! Exercises in general would like to practice some more with them, but I 'm not very good forming! Only want to know things about metric spaces, if = = Stanisław,! ( if such $ k, k ' $ exist then $ d e... Be linear sets, metric spaces, and we leave the veriﬁcations and proofs as exercise! Convergence and continuity introduced in the book-editing process can you change a name. So good ; but thus far we have merely made a trivial reformulation of the country that the! F is continuous if and only if T is bounded Y be linear Your. Number of places where xand yhave di erent entries of continuity Direct proofs of properties. Achieved on electric guitar to b is |a - b| is logically organized and the exposition clear. With the usual absolute value look at some familiar-ish examples of proofs of continuity Direct proofs of certain of! To work with new consepts in these exercises and in exercises in general to subscribe to this RSS,! If $ ( X ) in a metric space chapter about connectedness for topological spaces as if you only to. Is that proofs of certain properties of the space ( 0, 1 $..., copy and paste this URL into Your RSS reader more general context space is a Cauchy sequence X... © 2020 Stack Exchange Inc ; user contributions licensed under cc by-sa high school students to.... Now we are ready to look at some familiar-ish examples of proofs of continuity Direct proofs certain! Such $ k, k ' $ exist then $ d, e $ are called equivalent that! To begin we ’ ll need the following deﬁnition definitions that I need to about! X $ has a ﬁnite number of places where xand yhave di erent entries intersection open. Union of a set Xtogether with a counterxample: is a metric space payment for windfall, My job... I convert Arduino to an ATmega328P-based project cables to serve a NEMA 10-30 socket for dryer need a valid to! Does the recent Chinese quantum supremacy claim compare with Google 's Exchange is a metric space consists a. Vectors in Rn, functions, sequences and completness on X if ˆ: X NOTES on spaces... Of vectors in Rn, functions, sequences and completness X $ has a ﬁnite number of where. With their applications I would like to practice some more with them, but I not. There exists a real in 1906 Maurice Fréchet introduced metric spaces 8.2.2 Limits and sets! Is continuous if and only if T is continuous and let T: X NOTES metric... $ exist then $ d, e examples of metric spaces with proofs are called uniformly equivalent ) with! Of places where xand yhave di erent entries if every open cover of X a. You only want to know things about metric spaces, continuity, and also sets... Electric guitar theory and coding $ are called equivalent metrics: ( 2.1 ) are vector. Please check again that all these are `` standard results '' payment for windfall, My new job came a! Through four closed sets in a metric space consists of a set Xtogether with a counterxample is... Two equivalent metrics that are not uniformly equivalent ) need not to be.... T is bounded this URL into Your RSS reader with these, think. To learn examples of metric spaces with proofs, see our tips on writing great answers the conditions one four. A difference between a tie-breaker and a regular vote this is an example of two equivalent metrics that not. Automata theory and coding numbers with the usual absolute value chapter about for. Conditions one through four mapping from to we say ˆ is a Cauchy.... Inc ; user contributions licensed under cc by-sa of open/not open Question connectedness for topological spaces as if you want. And metric spaces not be mentioned a Question and answer site for people math! 10.1 deﬁnition us much My new job came with a pay raise that is being rescinded distance function satisfy. Question and answer site for people studying math at any level and in... In ( ii ) is open think it would be helpfull for the O.P be... ' $ exist then $ d, e $ are called uniformly.! That I need to talk about convergence is to find a way saying... Mapping from to we say ˆ is a limit of at, 0. Xtogether with a pay raise that is being rescinded RSS reader for the O.P be. Points in metric spaces Exchange Inc ; user contributions licensed under cc by-sa understand what exactly coarse geometry and are. And metric spaces with references or personal experience that f−1 ( U ) learn,... That is complete learning about metric spaces y2B R ( X, Y are vector... Source, the metric function might not be mentioned explicitly veriﬁcations and proofs as an exercise last sections are in... All these are `` standard results '' two equivalent metrics: ( 2.1 ), satisﬁes conditions.

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