Metric Spaces, Open Balls, and Limit Points DEFINITION: A set 푋, whose elements we shall call points, is said to be a metric space if with any two points 푝 and 푞 of 푋 there is associated a real number 푑(푝, 푞) called the distance from 푝 to 푞. This distance function 푑: 푋 × 푋 → ℝ must satisfy the following properties: (a) 푑(푝, 푞) > 0 if 푝 ≠ 푞 and 푑(푝, 푞) = 0 if 푝 = 푞; nonnegative property and zero property (b) 푑(푝, 푞) = 푑(푞, 푝); symmetric property (c) 푑(푝, 푞) ≤ 푑(푝, 푟) + 푑(푟, 푞), ∀푟 ∈ 푋. triangle inequality We denote a metric space by the ordered pair (푋, 푑). 2 EXAMPLE: Here are three different distance functions in ℝ . Let 풂 = (푥1, 푦1) and 풃 = (푥2, 푦2) be two points in ℝ2. 1. 푑1(풂, 풃) = |푥1 − 푥2| + |푦1 − 푦2| the taxicab metric 2 2 2. 푑2(풂, 풃) = √(푥1 − 푥2) + (푦1 − 푦2) the Euclidean metric (this is VERY familiar to you) 3. 푑∞(풂, 풃) = Max{ |푥1 − 푥2| , |푦1 − 푦2| } the infinity metric Consider a circle of fixed radius 푟. The picture on the right gives three different examples of a circle using the various distance functions. DO YOU SEE WHY? Remember, a circle of radius 푟 centered at the origin in the metric space (ℝ2, 푑) is the set of points {(푥, 푦) ∶ 푑((푥, 푦), (0,0)) = 푟} QUESTION: Which metric gives which “ball” and why? [You Do!] 2 2 The metric called 푑2(풂, 풃) = √(푥1 − 푥2) + (푦1 − 푦2) for 풂 = (푥1, 푦1) and 풃 = (푥2, 푦2) is the USUAL EUCLIDEAN DISTANCE FORMULA in ℝ2. MORE EXAMPLES: 1-dimensional Euclidean space (ℝ, 푑) QUESTION: What is the usual Euclidean distance formula on the real line? [You Do!] Spaces of Continuous Functions (퐶([0,1]), 푑) QUESTION: Does this satisfy all three properties of a metric? [You Do!] (just sketch main ideas) Spaces of Continuous Functions (퐶([0,1]), 푑) again QUESTION: Does this satisfy all three properties of a metric? [You Do!] (just sketch main ideas) The Discrete Metric EXAMPLE: Let E={New York City, Eau Claire, London, Beijing} QUESTION: Which city is furthest from Eau Claire? [You Do!] EXAMPLE: Let 퐸 = ℝ2 for example, the white/chalkboard. And let 푑 be the discrete metric. TASK: Rigorously prove that the space (ℝ2, 푑) is a metric space. [You Do!] Open (Closed) Balls in any Metric Space (푋, 푑) DEFINITION: Let 푋 be a space with metric 푑. Let 푥 ∈ 푋. Then the OPEN BALL of radius 푟 > 0 around 푥 is defined to be 푁푟(푥) ≔ {푦 ∈ 푋 | 푑(푥, 푦) < 푟}. QUESTION: What changes in the definition to make it a closed ball? [You Do!] TERMINOLOGY: The open ball 푁푟(푥) is called the neighborhood of radius 푟 about 푥. 2 TASK: Draw 푁2((0,2)) in the space ℝ with the usual Euclidean metric. [You Do!] TASK: Draw 푁2(1.5) in the space ℝ with the usual Euclidean metric. [You Do!] BOTTOM-LINE ABOUT OPEN BALLS IN EUCLIDEAN SPACE ℝ풏 WITH THE USUAL METRIC: In ℝ every open ball is just a ______________. In ℝ2 every open ball is just a ______________. In ℝ3 every open ball is just a ______________. In ℝ4 every open ball is just a ______________. Etc. 2 TASK: Draw 푁2((0,2)) and 푁1 ((0,2)) in the space ℝ with the discrete metric. [You Do!] ⁄2 TASK: Draw 푁2(1.5) and 푁1 (1.5) in the space ℝ with the discrete metric. [You Do!] ⁄2 BOTTOM-LINE ABOUT OPEN BALLS IN EUCLIDEAN SPACE ℝ풏 WITH THE DISCRETE METRIC: In ℝ every open ball is either a ________ OR _________. In ℝ2 every open ball is either a ________ OR _________. In ℝ3 every open ball is either a ________ OR _________. Limit Points in a metric space (푋, 푑) DEFINITION: Let 퐸 be a subset of metric space (푋, 푑). A point 푝 ∈ 푋 is a limit point of 퐸 if every neighborhood of 푝 contains a point 푞 ∈ 퐸 such that 푞 ≠ 푝. TASK: Write down the definition of “a point 푝 ∈ 푋 is NOT a limit point of 퐸”. [You Do!] QUESTION: If 푝 is a limit point, then does EVERY NEIGHBORHOOD of 푝 contain infinitely many points of 퐸? [We Will Do LATER and not now!] TWO RUNNING EXAMPLES (for notes in your notebook and not in this handout): 1 RUNNING EXAMPLE 1: Let 퐴 ⊂ ℝ be the set of points { | 푛 ∈ ℕ}. 푛 RUNNING EXAMPLE 2: Let 퐵 ⊂ ℝ2 be the set of points in the set {(푥, 푦) | 2 ≤ 푥 ≤ 6 푎푛푑 2 ≤ 푦 < 4} ∪ {(3,6), (5,6)} ∪ 푁1 ((4,5)) \ {(4,5)} ⁄2 TASK: Draw both of the sets above. [You Do!] .