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Topological Data Analysis: Assignment 1 Due Date: 19/08/2019

This document is a topological data analysis assignment with 6 problems: 1. Sketch unit spheres for p=1, 2, 3, and infinity in R2. 2. Determine all possible metrics on the set {+, -} and justify them. 3. Show that the function counting different entries is a metric on the set of all triples of 0s and 1s. 4. Show that two inequalities about distances in a metric space are true. 5. Determine if a function defines a metric on the power set of a metric space. 6. Determine if the closure of an open ball is always the closed ball and justify the answer.

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55 views1 page

Topological Data Analysis: Assignment 1 Due Date: 19/08/2019

This document is a topological data analysis assignment with 6 problems: 1. Sketch unit spheres for p=1, 2, 3, and infinity in R2. 2. Determine all possible metrics on the set {+, -} and justify them. 3. Show that the function counting different entries is a metric on the set of all triples of 0s and 1s. 4. Show that two inequalities about distances in a metric space are true. 5. Determine if a function defines a metric on the power set of a metric space. 6. Determine if the closure of an open ball is always the closed ball and justify the answer.

Uploaded by

ShubhamParashar
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Topological Data Analysis

Assignment 1
Due Date: 19/08/2019

Problem 1: For an 𝑛-dimensional Euclidean space ℝ𝑛 consider the following metrics:


1
∑𝑛 |𝑥 − 𝑦𝑖 |𝑝 ] 𝑝 , 1 ≤ 𝑖 < ∞;
𝑑𝑝 (x, y) ∶= [ 𝑖=1 𝑖
{max{|𝑥𝑖 − 𝑦𝑖 | ∣ 1 ≤ 𝑖 ≤ 𝑛}, 𝑝 = ∞.

Sketch unit spheres (i.e., the set of all points that are at a distance 1 from the origin) in 𝑅2 for
𝑝 = 1, 2, 3 and ∞. You may use the same coordinate axes to draw all the four spheres.
Problem 2: Let 𝑋 = {+, −} be the set of two distinct elements. Determine all possible metrics
on 𝑋. Provide some justification.
Problem 3: Let 𝐻 be the set of all triples of 0’s and 1’s. For 𝑥, 𝑦 ∈ 𝐻, show that the function
𝑑(𝑥, 𝑦) defined as the number of places where 𝑥 and 𝑦 have different entries is a metric.
Problem 4: Let (𝑋, 𝑑) be a metric space and 𝑥, 𝑦, 𝑧, 𝑤 ∈ 𝑋. Show that the following two
inequalities are true.
1. |𝑑(𝑥, 𝑦) − 𝑑(𝑧, 𝑤)| ≤ 𝑑(𝑥, 𝑧) + 𝑑(𝑦, 𝑤).

2. |𝑑(𝑥, 𝑧) − 𝑑(𝑦, 𝑧)| ≤ 𝑑(𝑥, 𝑦).

Problem 5: For a metric space (𝑋, 𝑑) let 𝒫 (𝑋) denote the power set of 𝑋(i.e., set of all subsets
of 𝑋). Consider the following function on 𝒫 (𝑋) × 𝒫 (𝑋) -

𝒟 (𝐴, 𝐵) ∶= inf{𝑑(𝑎, 𝑏) ∣ 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵}.

Does the above function define a metric on 𝒫 (𝑋)? Justify your answer.
Problem 6: Recall that an open ball, 𝐵𝑟 (𝑎), of radius 𝑟 around a point 𝑎 is the set of all points
that are at a distance at most 𝑟 from 𝑎. The closed ball around 𝑎 of radius 𝑟, denoted 𝐵 𝑟 (𝑎), is the
union of open ball of radius 𝑟 and the points that are at a distance exactly 𝑟 from 𝑎. Is it always
true that the closure of 𝐵𝑟 (𝑎) must be 𝐵 𝑟 (𝑎)? Justify your answer.

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