(2) Let $(X, d)$ be a metric space and $B_d(x, \epsilon) = \{y \in X \mid d(x, y) < \epsilon\}$. Show that $B = \{B_d(x, \epsilon) \mid x \in X, \epsilon > 0\}$ is a basis for a topology on $X$ which is called the metric topology $\mathcal{T}_d$.
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For any x ∈ X, there exists a B ∈ B such that x ∈ B. 2. If B1, B2 ∈ B and x ∈ B1 ∩ B2, then there exists a B3 ∈ B such that x ∈ B3 ⊆ B1 ∩ B2. Show more…
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Theorem 9.8. A metric space is Hausdorff, regular, and normal. Definition. A metric on a set M is a function d : M × M → ℝ₊ (where ℝ₊ is the non-negative real numbers) such that for all a, b, c ∈ M, these properties hold: (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 definite, 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 ∈ M, d(a, b) = 1 if a ≠ b and d(a, a) = 0. This metric basically tells us whether two points are the same or different. Example. Here’s a strange metric on ℚ: for reduced fractions, let d(a/b, m/n) = max(|a − m|, |b − n|). Which rationals are “close” to one another under this metric? Theorem 9.3. Let d be a metric on the set X. Then the collection of all open balls ℬ = {B(p, ϵ) = {y ∈ X|d(p, y) < ϵ} for every p ∈ X and every ϵ > 0} forms a basis for a topology on X. The topology generated by a metric d on X is called the d-metric topology for X. Definition. A topological space (X, ጒ) is a metric space or is metrizable if and only if there is a metric d on X such that ጒ is the d-metric topology. We sometimes write a metric space as (X, d) to denote X with the d-metric topology. Theorem 9.6. For any metric space (X, d), there exists a metric d̄ such that d and d̄ generate the same topology, yet for each x, y ∈ X, d̄(x, y) < 1. Theorem 9.7. If X is a metric space and Y ⊂ X, then Y is a metric space.
Adi S.
(a) Show that every separable metric space has a countable base (b) Show that any compact metric space K has a countable base, and that K is therefore separable
Urvan J.
Let X be a metric space with metric d. (a) Show that d : X x X → R is continuous. (b) Let X' denote a space having the same underlying set as X. Show that if d : X' x X' → R is continuous, then the topology of X' is finer than the topology of X. One can summarize the result of this exercise as follows: If X has a metric d, then the topology induced by d is the coarsest topology relative to which the function d is continuous.
Sri K.
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