Theorem 9.3. Let d be a metric on the set X. Then the collection of all open balls B = {B(p, e) = {y E X\d(p, y) < e} for every p E X and every e > 0} forms a basis for a topology on X.
Theorem 9.3. Let d be a metric on the set X. Then the collection of all open balls B = {B(p, e) = {y E X\d(p, y) < e} for every p E X and every e > 0} forms a basis for a topology on X.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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How do i prove 9.3? Could you include step by step explanation? I included a question and some related definitions in my textbook.
![Theorem 9.3. Let d be a metric on the set X. Then the collection of all open balls
B
(В (р, е) 3 {у € X/а(р, у) < e} for every p € X аnd every є > 0}
forms a basis for a topology on X.
Definition. A metric on a set M is a function d : M × M
non-negative real numbers) such that for all a, b, c E 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.
Example. On any set M, we can define the discrete metric as follows: for any a, b e
М, d(а, b)
points are the same or different.
1 if a + b and d(a, a)
= 0. This metric basically tells us whether two
m
Example. Here's a strange metric on Q: for reduced fractions, let d(÷, “) = max(|a –
b’ n
m], |b – n|). Which rationals are "close" to one another under this metric?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F9ef60767-66d0-442b-9965-f03831f73e0d%2Fhn5eah_processed.png&w=3840&q=75)
Transcribed Image Text:Theorem 9.3. Let d be a metric on the set X. Then the collection of all open balls
B
(В (р, е) 3 {у € X/а(р, у) < e} for every p € X аnd every є > 0}
forms a basis for a topology on X.
Definition. A metric on a set M is a function d : M × M
non-negative real numbers) such that for all a, b, c E 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.
Example. On any set M, we can define the discrete metric as follows: for any a, b e
М, d(а, b)
points are the same or different.
1 if a + b and d(a, a)
= 0. This metric basically tells us whether two
m
Example. Here's a strange metric on Q: for reduced fractions, let d(÷, “) = max(|a –
b’ n
m], |b – n|). Which rationals are "close" to one another under this metric?
![Definition. Let I be a topology on a set X, and let B C J. Then B is a basis for the
topology T if and only if every open set in T is the union of elements of B. If B E B,
we say B is a basis element or basic open set. Note that B is an element of the basis
B, but a subset of the space X.
Theorem 3.1. Let (X,J) be a topological space, and let B be a collection of subsets of X.
Then B is a basis for T if and only if
(1) В С Т, апd
(2) for each set U in T and point p in U there is a set V in B such that p E V C U.
Theorem 3.3. Suppose X is a set and B is a collection of subsets of X. Then B is a basis
for some topology on X if and only if
(1) each point of X is in some element of B, and
(2) if U and V are sets in B and p is a point in U n V, there is a set W in B such that
peW C (Un V).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F9ef60767-66d0-442b-9965-f03831f73e0d%2Frqmo5n4_processed.png&w=3840&q=75)
Transcribed Image Text:Definition. Let I be a topology on a set X, and let B C J. Then B is a basis for the
topology T if and only if every open set in T is the union of elements of B. If B E B,
we say B is a basis element or basic open set. Note that B is an element of the basis
B, but a subset of the space X.
Theorem 3.1. Let (X,J) be a topological space, and let B be a collection of subsets of X.
Then B is a basis for T if and only if
(1) В С Т, апd
(2) for each set U in T and point p in U there is a set V in B such that p E V C U.
Theorem 3.3. Suppose X is a set and B is a collection of subsets of X. Then B is a basis
for some topology on X if and only if
(1) each point of X is in some element of B, and
(2) if U and V are sets in B and p is a point in U n V, there is a set W in B such that
peW C (Un V).
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