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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

Transcribed Image Text:**Title: Understanding Open Sets in Metric Spaces**
**Text:**
"Show that an open ball in a metric space is an open set."
**Explanation:**
In the context of metric spaces, an open ball is a fundamental concept used to describe neighborhoods around a point. An open ball centered at a point \( x \) with radius \( r \) is the set of all points whose distance from \( x \) is less than \( r \). To prove that an open ball is an open set, we show that for any point inside the ball, we can find a smaller ball entirely contained within the larger ball. This ensures that the point is an interior point, satisfying the definition of an open set.
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I think there is a mistake in the last part. It says B epsilon (x) subset =B r(x). It supposed to be B r(a). Please let me know. Thank you.
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