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
Related questions
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
Transcribed Image Text: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
Transcribed Image Text: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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