Question

4. Let $X = \mathbb{R}$ and $d(x, y) = |x - y|$. Show that an interval $(a, b)$ is open for any $a, b \in \mathbb{R}$ with $a < b$.

Name: 4 let x mathbbr and dx y x y show that an interval a b is open for any a b in mathbbr with a b 88077
Uploaded: 2022-01-03T22:03:32-08:00
Duration: 3 min 28 s
Channel: Apratim De
Description: 4 let x mathbbr and dx y x y show that an interval a b is open for any a b in mathbbr with a b 88077

          4. Let $X = \mathbb{R}$ and $d(x, y) = |x - y|$. Show that an interval $(a, b)$ is open for any $a, b \in \mathbb{R}$ with $a < b$.

Added by Francisco Javier M.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Apratim De

01/03/2022

Step 1

We want to show that $(a, b)$ is an open set. By definition, a set is open if for every point $x$ in the set, there exists an $\epsilon > 0$ such that the open ball $B(x, \epsilon) = \{y \in \mathbb{R} : d(x, y) < \epsilon\}$ is contained in the set. Show more…

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