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Let X = R, and let T consist of 0, R, and all intervals of the form (a, b) where a, b e R and a < b. Show that T is not a topology on IR. %3D

Let X = R, and let T consist of 0, R, and all intervals of the form (a, b) where a, b e R and a < b. Show that T is not a topology on IR. %3D

Advanced Engineering Mathematics
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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Prove that the first example is a topology and the second is not.

Let \( X = \mathbb{R} \), and let \( \mathcal{T} \) consist of \(\emptyset, \mathbb{R}\), and all intervals of the form \((a, b)\) where \(a, b \in \mathbb{R}\) and \(a < b\). Show that \(\mathcal{T}\) is not a topology on \(\mathbb{R}\).
Transcribed Image Text:Let \( X = \mathbb{R} \), and let \( \mathcal{T} \) consist of \(\emptyset, \mathbb{R}\), and all intervals of the form \((a, b)\) where \(a, b \in \mathbb{R}\) and \(a < b\). Show that \(\mathcal{T}\) is not a topology on \(\mathbb{R}\).
Let \( X = \mathbb{N} \) and let \( \mathcal{T} \) consist of \(\emptyset, \mathbb{N}\), and all sets \( A_n = \{ n, n + 1, n + 2, \ldots \} \) where \( n \geq 1 \). 1. Show that this is a topology on \(\mathbb{N}\).
Transcribed Image Text:Let \( X = \mathbb{N} \) and let \( \mathcal{T} \) consist of \(\emptyset, \mathbb{N}\), and all sets \( A_n = \{ n, n + 1, n + 2, \ldots \} \) where \( n \geq 1 \). 1. Show that this is a topology on \(\mathbb{N}\).
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