Topology: Basic Pseudometric Spaces. Please check Q4 and the solution I had for part c, was told 4c is not correct. The closure of S is not the open interval (-1,1) on the x_1-axis. It is a vertical strip in the plane above and below the interval [-1,1] on the x_1-axis. Kindly give a better solution الف1st Condition2nd Condition is Satisfied because ofThenboth X, andare-XIequal and the absolute value is equal to o3rd Condition is safisfied because |X₁-y₁ = y₁=x₁o(x,is satisfied because o(X₁2) = 1×₁-2₁o(x,y) + 0 (y₁z)4th6. The 8- Celldefined,XI-5=C-lo: 0(x,y) = (x²-y₁) is a p.m on X.メーBRSANTadiusofas the Sefآےilof£x = ¹2-1xfind the o-closure of S= {XEX: X1^2 + X2^2 <1},need to find the Set ofallpoints (X1, X₂) in x sach thatevery >there exists a point (X₁, X2) in S such thettXI-X₁ 1. Let X be the set of real numbers with the usual pseudometric. Find thederived set and the closure of each of the following sets:(a) A = {1/n: n = 1,2,3,...);(b)(c)B= Z, the integers;C= Q, the rational numbers.2. Let Xbe the reals and define p(x, y) = 2x - y). Show that p is a pseudo-metric equivalent to the usual pseudometric for X.3. Let X be the plane and for x =1-₁ +232|-(₁, 2) and y = (1, 32), let p(x, y) =(a) Show that p is a pseudometric.(b) Describe the p-cell of radius r centered at the point (a, b).(e) Find the p-closure of S = {re X: r + x² <1}.4. Let X be the plane. For r = (₁, 2) and y = (v₁.32) in X, let o(z. y) =|-yil.(a) Show that is a pseudometric.(b) Describe the o-cell of radius r centered at the point (a, b).(e) Find the o-closure of S = {z € X: +z<1}.5. (a) Is the pseudometric p in Exercise 3 equivalent to the usual pseudometricfor the plane? Explain.

It is given that X is a plane with x=x1,x2, y=y1,y2 and x,y∈X, σx,y=x1-y1. The objective is to…

4. Let X be the plane. For z = (₁, 2) and y = (1.32) in X, let a(x, y) = 21-yil. (a) Show that is a pseudometric. (b) Describe the d-cell of radius r centered at the point (a, b). (e) Find the d-closure of S= {re X: z}+x} <1}.

4. Let X be the plane. For z = (₁, 2) and y = (1.32) in X, let a(x, y) = 21-yil. (a) Show that is a pseudometric. (b) Describe the d-cell of radius r centered at the point (a, b). (e) Find the d-closure of S= {re X: z}+x} <1}.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Topology: Basic Pseudometric Spaces. Please check Q4 and the solution I had for part c, was told 4c is not correct. The closure of S is not the open interval (-1,1) on the x_1-axis. It is a vertical strip in the plane above and below the interval [-1,1] on the x_1-axis. Kindly give a better solution

الف
1st Condition
2nd Condition is Satisfied because of
Then
both X, and
are
-XI
equal and the absolute value is equal to o
3rd Condition is safisfied because |X₁-y₁ = y₁=x₁
o(x,
is satisfied because o(X₁2) = 1×₁-2₁
o(x,y) + 0 (y₁z)
4th
6. The 8- Cell
defined,
XI-
5
=
C-lo
: 0(x,y) = (x²-y₁) is a p.m on X.
メー
BR
SAN
Tadius
of
as the Sef
آے
il
of
£
x = ¹2-1x
find the o-closure of S= {XEX: X1^2 + X2^2 <1},
need to find the Set of
all
points (X1, X₂) in x sach that
every >
there exists a point (X₁, X2) in S such thett
XI-X₁ <T. Since the P.m function in this case only
Since the P.m. function
S
on the X₁ Coordinate, the o-closure of I could
depends
be the set of all points (X₁, X2) in X such that X1^241
which is an
interval (-1, 1) on the XI axis
open
|xı-y₁l+y₁=zı
all
I Centered at point (a, b) can be
points (X₁, X₂) in X such that
ix
چھوڑا

1. Let X be the set of real numbers with the usual pseudometric. Find the
derived set and the closure of each of the following sets:
(a) A = {1/n: n = 1,2,3,...);
(b)
(c)
B= Z, the integers;
C= Q, the rational numbers.
2. Let X
be the reals and define p(x, y) = 2x - y). Show that p is a pseudo-
metric equivalent to the usual pseudometric for X.
3. Let X be the plane and for x =
1-₁ +232|-
(₁, 2) and y = (1, 32), let p(x, y) =
(a) Show that p is a pseudometric.
(b) Describe the p-cell of radius r centered at the point (a, b).
(e) Find the p-closure of S = {re X: r + x² <1}.
4. Let X be the plane. For r = (₁, 2) and y = (v₁.32) in X, let o(z. y) =
|-yil.
(a) Show that is a pseudometric.
(b) Describe the o-cell of radius r centered at the point (a, b).
(e) Find the o-closure of S = {z € X: +z<1}.
5. (a) Is the pseudometric p in Exercise 3 equivalent to the usual pseudometric
for the plane? Explain.

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