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Let $X$ be a set and assume that p E X. Let Tp be the collection 0, X, and all subsets of X that exclude n

Let $X$ be a set and assume that p E X. Let Tp be the collection 0, X, and all subsets of X that exclude n

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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**Statement:** Show that \( \mathcal{T}_p \) is a topology on \( X \). --- **Explanation:** To establish that \( \mathcal{T}_p \) is a topology on the set \( X \), you must verify the following three conditions: 1. **The Empty Set and \( X \):** Both the empty set \( \emptyset \) and the set \( X \) must be in \( \mathcal{T}_p \). 2. **Closed Under Arbitrary Unions:** The collection \( \mathcal{T}_p \) must be closed under arbitrary unions. This means that for any collection of sets \(\{U_i\}\) where each \(U_i\) is in \( \mathcal{T}_p \), the union \(\bigcup U_i\) must also be in \( \mathcal{T}_p \). 3. **Closed Under Finite Intersections:** The collection \( \mathcal{T}_p \) must be closed under finite intersections. This implies that for any finite collection of sets \(\{U_1, U_2, \ldots, U_n\}\) where each \(U_i\) is in \( \mathcal{T}_p \), the intersection \(\bigcap_{i=1}^n U_i\) must also be in \( \mathcal{T}_p \). These conditions ensure that \( \mathcal{T}_p \) is indeed a topology on \( X \).
Transcribed Image Text:**Statement:** Show that \( \mathcal{T}_p \) is a topology on \( X \). --- **Explanation:** To establish that \( \mathcal{T}_p \) is a topology on the set \( X \), you must verify the following three conditions: 1. **The Empty Set and \( X \):** Both the empty set \( \emptyset \) and the set \( X \) must be in \( \mathcal{T}_p \). 2. **Closed Under Arbitrary Unions:** The collection \( \mathcal{T}_p \) must be closed under arbitrary unions. This means that for any collection of sets \(\{U_i\}\) where each \(U_i\) is in \( \mathcal{T}_p \), the union \(\bigcup U_i\) must also be in \( \mathcal{T}_p \). 3. **Closed Under Finite Intersections:** The collection \( \mathcal{T}_p \) must be closed under finite intersections. This implies that for any finite collection of sets \(\{U_1, U_2, \ldots, U_n\}\) where each \(U_i\) is in \( \mathcal{T}_p \), the intersection \(\bigcap_{i=1}^n U_i\) must also be in \( \mathcal{T}_p \). These conditions ensure that \( \mathcal{T}_p \) is indeed a topology on \( X \).
Let \( X \) be a set and assume that \( p \in X \). Let \( \mathcal{T}_p \) be the collection \(\emptyset, X\), and all subsets of \( X \) that exclude \( p \).
Transcribed Image Text:Let \( X \) be a set and assume that \( p \in X \). Let \( \mathcal{T}_p \) be the collection \(\emptyset, X\), and all subsets of \( X \) that exclude \( p \).
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