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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
Advanced Engineering Mathematics
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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?
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).
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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