#1 Exercise 1. Let X be a metric space. The subsets of X which are arbitrary(finite or infinite) unions of open balls form the open sets of a topology onX.

Answered: Exercise 1. Let X be a metric space.…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Exercise 1. Let X be a metric space. The subsets of X which are arbitrary
(finite or infinite) unions of open balls form the open sets of a topology on
X.

Expert Solution

Step 1

What is Open Set:

Open sets in a metric space are those that, for every point P, include all other points that are sufficiently close to P. Open sets are more broadly defined as the constituents of a given collection of subsets of a given set, a collection that has the property of including the entire set, the empty set, every union of its constituents, and every finite intersection of its constituents. A topological space is a set in which such a collection is given, and a topology is the name of the collection.

To Prove:

We prove that in a metric space $X$ , every subsets of $X$ that are arbitrary unions of open balls form the open set of topology in $X$ .