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Handout on bases of topologies Definition 1. Let X be a nonempty set and T be a topology on X. A family B of subsets of X is called a basis for T or a base for T if (1) For every B ∈ B we have B ∈ T and (2) Every nonempty set U ∈ T can be represented as a union of some family of sets from B. If B is a basis for a topology T on X, we also say that B generates T. Theorem 1 [Criterion for being a basis of T ] Let X be a nonempty set, T be a topology on X and let B be a family of sets from T . Then B is a basis for T if and only if whenever U ∈ T and x ∈ U , then there exists B ∈ B such that x ∈ B ⊆ U . Example 1. (1) {(a, b)|a, b ∈ R, a < b} is a basis for the Euclidean topology Te on R. (2) {(a, b)|a, b ∈ Q, a < b} is a basis for the Euclidean topology Te on R. (3) {(−∞, a)|a ∈ R} is not a basis for the Euclidean topology Te on R. (4) {(a, b) × (c, d)|a, b, c, d ∈ Q, a < b, c < d} is a basis for the Euclidean topology Te on R2 . (5) For every integer m ≥ 1 the family {B(x, n1 )|x ∈ Rm , n ≥ 1, n ∈ Z} is a basis for the Euclidean topology Te on Rm . (6) Let X be a nonempty set. Then {{x}|x ∈ X} is a basis for the discrete topology on X. (7) Let X be a nonempty set. Then {X} is a basis for the trivial topology on X. (8) Let X = R. Then {(a, b)|a, b ∈ R, a < b} is not a basis for the trivial topology on X. (9) Let X = R. For a ∈ R let Ua := R \ {a} = (−∞, a) ∪ (a, ∞). Then {Ua |a ∈ R} is not a basis for the finite-complement topology Tf c on R. Definition 2. Let X be a nonempty set. A family B of subsets of X is called a topological basis with respect to X or a topological basis on X if the following two conditions hold: (1) For every x ∈ X there exists B ∈ B such that x ∈ B. 1 2 (2) Whenever x ∈ X, B1 , B2 ∈ B are such that x ∈ B1 ∩ B2 then there exists B ∈ B such that x ∈ B ⊆ B1 ∩ B2 . Theorem 2. Let X be a nonempty set and let B be a family of subsets of X. Then the following hold: (1) B is a basis for some topology on X if and only if B is a topological basis with respect to X. (2) If B is a topological basis with respect to X, then there exists a unique topology T on X such that B is a basis for T . Namely, T consists of the empty set and of all nonempty subsets U of X such that U represented as the union of some family of sets from B. Example 2: (1) Let X = R and let B = {(−∞, a)|a ∈ R} ∪ {(b, ∞)|b ∈ R}. Then B is not a topological basis with respect to X since property (2) from Definition 2 does not hold for B. Therefore there does not exist a topology on X such that B is a basis for this topology. (2) Let X = R and B = {(a, b] |a, b ∈ R, a < b}. Then B is a topological basis with respect to X. Therefore there exists a unique topology T on R such that B is a basis for T . This topology is called the upper limit topology on R. (3) For each n ∈ Z define the set A(n) as follows: A(n) = {n} if n is odd and A(n) = {n − 1, n, n + 1} if n is even. Put B = {A(n)|n ∈ Z}. Then B is a topological basis on Z. Therefore there exists a unique topology T on Z such that B is a basis for T . This T is called the digital topology on Z. (4) Let X be a nonempty set and let F be the family of all finite subsets of X. Determine whether or not F is a topological basis on X. If F is a topological basis on X, determine which topology F generates. (5) Let X, Y be nonempty sets. Let T1 be a topology on X and let T2 be a topology on Y . Put B = {U × V |U ∈ T1 , V ∈ T2 } Then B is a topological basis on X ×Y . The topology on X ×Y generated by B is called the product topology. 3 Think about whether or not B itself is a topology on X × Y . (6) Let X = R and let Y ⊆ X. Determine for which Y the family {Y } is a topological basis on R. (7) Let X = R and B = {[a, b] : a, b ∈ R, a < b} Is B a topological basis on R?