Could you explain how to show 9.7 in detail? Theorem 9.7. If X is a metric space and Y C X, then Y is a metric space.Definition. A metric on a set M is a function d : M x M →non-negative real numbers) such that for all a, b, cE M, these properties hold:R. (where R, is the(1) d(a, b) > 0, with d(a, b) = 0 if and only if a = b;(2) d(a, b) = d(b, a);(3) d(a, c) < d(a, b) + d(b, c).These three properties are often summarized by saying that a metric is positive defi-nite, symmetric, and satisfies the triangle inequality.A metric space (M, d) is a set M with a metric d.Example. The function d(x, y) = |x – y| is a metric on R. This measure of distance isthe standard metric on R.%3DExample. On any set M, we can define the discrete metric as follows: for any a, b eM, d(a, b) = 1 if a + b and d(a, a) = 0. This metric basically tells us whether twopoints are the same or different.mExample. Here's a strange metric on Q: for reduced fractions, let d(÷, “) = max(|a –nm|, |b – n|). Which rationals are "close" to one another under this metric?Theorem 9.3. Let d be a metric on the set X. Then the collection of all open ballsВ %3 {В(р, є) — {y € X\d(p, у) < e} for every p ЕХаnd every e> 0}forms a basis for a topology on X.The topology generated by a metric d on X is called the d-metric topology for X.Definition. A topological space (X, T) is a metric space or is metrizable if and onlyif there is a metric d on X such that J is the d-metric topology. We sometimes write ametric space as (X, d) to denote X with the d-metric topology.Theorem 9.6. For any metric space (X, d), there exists a metric d such that d and dgenerate the same topology, yet for each x, y E X, d(x,y) < 1.

Name: Could you explain how to show 9.7 in detail? Theorem 9.7. If X is a me
Uploaded: 2024-03-20T06:46:03.000Z
Channel: Bartleby
Description: Could you explain how to show 9.7 in detail? Theorem 9.7. If X is a metric space and Y C X, then Y is a metric space.Definition. A metric on a set M is a function d : M x M →non-negative real numbers) such that for all a, b, cE M, these properties ho

Answered: Image /qna-images/answer/d68c6e9a-1300-48f8-a500-3fc3555207b4.jpg

Answered: If X is a metric space and Y C X, then…

Math

Advanced Math

If X is a metric space and Y C X, then Y is a metric space.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...

See similar textbooks

Related questions

Q: 1- a = (1, .,0) in the usual metric space R. 2- (0,1) in the usual metric space R. 3- {(x, y): xy =…

A: As per our company guidelines we are supposed to answer ?️only first 3 sub-parts. Kindly repost…

Q: Is the euclidean space R³ is a complete metric space

A: We have to check whether euclidean space R³ is a complete metric space or not .

Q: Let (X, d) be a metric space and let A ⊆ X be a compact subset. show that, If B ⊆ X and d(A, B) =…

Q: Question 7. Let (X,r) be a topological space and A, BCX, Determine whether A-B2A-B or not.

Q: If X =R and d(x.y)30, then X metric space not metric semi metric

Q: (b) Is (R2,d) a metric space, where d((x.y1).(x2.y2)) 1x1-x2l; for all (x, , y1 ), (x2 2) E R2 ?…

Q: Let (R. d) be discrete metric space then R is not complete O True O Faise

Q: Example (4): Let (M, d) be a metric space. Define a function e : M X M → R as: e(x,y) d(x,y)…

A: Metric space is the base of topology. A metric space is a function that maps a non-empty set under…

Q: Let X = [0, 1] and d be the usual metric, d(x, y) = |x- y| %3D O True O False

A: TRUE

Q: State whether the following is true or false: a) The set [0,4] with the usual metric is…

A: (a) The set [0,4] with the usual metric is sequentially compact. This statement is true. Considering…

Q: - Let (X, t) be a topological space and A, BCX, then 1- A° 2 A- b(A) 2- A° # A- b(A) 3- A° = A –…

Q: Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B)…

Q: (a) Suppose that (M, d) is a metric space. Consider the mapping f: M [0, 1] be given and let di(x,…

Q: Let X = R, und d be the discrete metric, und A = (3,5) Show that if A is compact or not

Q: Let R be the set of real numbers and d: R x R → R be defined as follows. d(x, y) = 5|x10 - y1º|.…

Q: Let (x,d) be metric space and A, BC X. 2. Show that (AnB)= A'n B and AUB = AUB %3D

Q: = 0,1] and d be the usual metric, d(x, y) = a -y %3D en d can not induces a norm on X True False

A: the usual metric d(x,y)=|x-y| usually induces the norm.

Q: Suppose Show that p: X²> t0,00) defined by d is a metric n X, 2. 3d (x,y) 2+3d (xy) metric an p…

A: Given that d is a metric on X. We need to prove that ρ is a metric.

Q: Let (M, d) be a metric space. Show that the set {B(x, 1/n); x E M; n e N*} is a Basis for the metric…

Q: Let (X, d) be a separable metric space. into l∞.

A: This is a problem of functional analysis.

Q: Consider IR as a metric space.with metric dux, y) = 1x-y1. Show that the open interval C0, 1) is an…

Q: Let (X, d) be a metric space, and p: X→ IR doxy) I+desy) de fined by puM pusy) = %3D show that p is…

A: Consider a set X and a function defined from it to the real numbers as d : X×X→ℝ. Then, X,d is a…

Q: If X =R and d(x,y)=0, then X metric but not semi metric O metric space and semi metric O not metric…

Q: Suppose (X, d) is a metric space and let r> 0. Show that p: X2 → [0, c0) defined by p(x, y) = r·…

Q: Example (3): Let (M, d) be a metric space. Define a function e: Mx M→R by: e(x,y) = Min{1,d(x y)};…

Q: (f) The set B = {(x, y) E R² : x² + y² < 1} as a subset of R? with indiscrete metric is…

Q: 2. Let (X, d) be a metric space and y E X. (i) Prove that the closed ball B[y,2] is a closed set in…

Q: Question 6. Let (X,r) be a topological space and A, BCX. Determine whether (A-B)CA -B or not.

Q: Theorem 9.13. Let (X, d) and (Y,e) be metric spaces. Then X × Y is a metric space.

Q: Let (C[a,b] .d) be the discrete metric space. Then Cla,b] is bounded. * False True

Q: A of a metric space (x, d.)

Q: Theorem 9.6. For any metric space (X, d), there exists a metric d such that d and d generate the…

A: Let X, d be any metric space. It is required to prove that there exists a metric d such that d and…

Q: For every ,x, y E R, let p(x, y) =| x² – y² |. Is p a metric -

Q: d(r, y) Let (X, d) be a metric space and let p(r, y) = for all r, y E X. Show that (X, d) 1+d(x, y)…

A: According to our company policy we are supposed to answer only first question i.e. (4) 4) in this…

Q: Let X and Y be topological spaces. Prove that X x Y is completely regular if and only if X and Y are…

Q: Let X=R and T = {]a, +oo[/a R}U{o,R}, then (X, 7) is a regular space. True False

A: This is a problem of topology.

Q: Prove that Let X be a metric space such that X E1 U E2. If E1, E2 are complete, then X is complete.

Q: QUESTION 3 Any cover of a metric space is open. O True O False

Q: Show that (C(X),|| · ||∞,x) is a metric space.

Q: (a) Consider the metric space 0 (i) if r 1.

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Let X and Y be metric spaces and f : X Y be continuous. Let S C X. now: (a) f(S) C f(S). (b) If S is…

A: Let X and Y be metric spaces.

Q: 12. Show that d((x1, Y1), (x2, Y2)) = |x1x2- Y1Y2| is not a metric on X = R %3D %3D

Q: Question 6. Let (X,r) be a topological space and A, BCX. Determine whether (A-B)' C A' – B* or not.

Q: Let (X, d) be a compact metric space and T: XX be a mapping that satisfies d(Tr,Ty)<d(r,y) for all…

A: What is Metric Space: Metric space is an abstract set in mathematics, particularly topology, with a…

Q: Theorem 9.8. A metric space is Hausdorff, regular, and normal.

A: The objective is to show that every metric is hausdorff, regular and normal. Let x≠y be points of a…

Q: if x is a compact Metric space and F e C (X₂Y) is equicontinuous, then F i's uniformly…

Q: Prove that: 1. (R, J. |) is a complete metric space. 2. (Q, I. |) is not complete metric

A: To prove that R,. is a complete metric space. Suppose that xn is increasing and bounded, Let…

Q: Exercise 1.33. Let (X,d) be a metric spáce and d(x,y) 1+d(x,y)' d* (x,y) = Prove that the family of…

A: This is problem from my old works, where I used e instead of d*.

Concept explainers

Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

Numbers

Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.

Subtraction

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

Addition

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.

Topic Video

Question

100%

Could you explain how to show 9.7 in detail?

Theorem 9.7. If X is a metric space and Y C X, then Y is a metric space.
Definition. A metric on a set M is a function d : M x M →
non-negative real numbers) such that for all a, b, cE M, these properties hold:
R. (where R, is the
(1) d(a, b) > 0, with d(a, b) = 0 if and only if a = b;
(2) d(a, b) = d(b, a);
(3) d(a, c) < d(a, b) + d(b, c).
These three properties are often summarized by saying that a metric is positive defi-
nite, symmetric, and satisfies the triangle inequality.
A metric space (M, d) is a set M with a metric d.
Example. The function d(x, y) = |x – y| is a metric on R. This measure of distance is
the standard metric on R.
%3D
Example. On any set M, we can define the discrete metric as follows: for any a, b e
M, d(a, b) = 1 if a + b and d(a, a) = 0. This metric basically tells us whether two
points are the same or different.
m
Example. Here's a strange metric on Q: for reduced fractions, let d(÷, “) = max(|a –
n
m|, |b – n|). Which rationals are "close" to one another under this metric?
Theorem 9.3. Let d be a metric on the set X. Then the collection of all open balls
В %3 {В(р, є) — {y € X\d(p, у) < e} for every p ЕХаnd every e> 0}
forms a basis for a topology on X.
The topology generated by a metric d on X is called the d-metric topology for X.
Definition. A topological space (X, T) is a metric space or is metrizable if and only
if there is a metric d on X such that J is the d-metric topology. We sometimes write a
metric space as (X, d) to denote X with the d-metric topology.
Theorem 9.6. For any metric space (X, d), there exists a metric d such that d and d
generate the same topology, yet for each x, y E X, d(x,y) < 1.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Algebraic Operations$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .
10. Let and be mappings from to. Prove that if is invertible, then is onto and is one-to-one.
If x and y are elements of an ordered integral domain D, prove the following inequalities. a. x22xy+y20 b. x2+y2xy c. x2+y2xy
For each of the following parts, give an example of a mapping from E to E that satisfies the given conditions. a. one-to-one and onto b. one-to-one and not onto c. onto and not one-to-one d. not one-to-one and not onto
Let (X,d) be a metric space , x ϵ X and A ⊑ X be a nonempy set. Prove that d (x ,A) = 0 if and only if every neighborhood of x contains a point of A.
Let (X, T ) be a topological space, (M, d) be a complete metric space andBC(X, M) := {f ∈ C(X, M); f[X] is bounded }d∞(f, g) := sup d(f(x), g(x)) (f, g ∈ BC(X, M)).Then (BC(X, M), d∞) is a complete metric space.
Let (X,d) be a metric space. For x,y in X define e(x,y)=min{1,d(x,y)}. Prove that (X,e) is also a metric space.
Let (X, d) be a metric space and let A ⊆ X be a compact subset. show that, If B ⊆ X and d(A, B) = inf{d(a, b) : a ∈ A and b ∈ B} = 0 then A ∩ B = ∅.
Let (X,d) be a metric space and d*(x,y) = d(x,y)/1+d(x,y) Prove that the family of open sets with respect to the metric d is same as the family of open sets with respect to the metric d*.
Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric.Hint: As usual, there are two directions to prove in an ⇔. The picture on p73 of the notes may be somewhat helpful.
Prove that if X,d is a metric space and G ⊆ H ⊆ X, then G-bar ⊆ H-bar.