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Show that in the finite complement topology on R every subspace is compact. Is R compact in the lower limit topology?

Show that in the finite complement topology on R every subspace is compact. Is R compact in the lower limit topology?

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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**Title: Compactness in Topologies** **Problem Statement:** Show that in the finite complement topology on \( \mathbb{R} \), every subspace is compact. Is \( \mathbb{R} \) compact in the lower limit topology? **Concepts to Explore:** 1. **Finite Complement Topology:** - In this topology, the open sets are those whose complements in \( \mathbb{R} \) are finite, along with the empty set. 2. **Compactness:** - A subspace is compact if every open cover has a finite subcover. 3. **Lower Limit Topology:** - In this topology, open intervals are of the form \([a, b)\). **Discussion:** - **Compactness in Finite Complement Topology:** - Demonstrate why every subspace in this topology is compact by considering the nature of open sets and their complementarities. - **Compactness in Lower Limit Topology:** - Investigate whether \( \mathbb{R} \) with this topology is compact, considering the structure of the open intervals. Explore these concepts to deepen your understanding of topology and compactness properties within different topological spaces.
Transcribed Image Text:**Title: Compactness in Topologies** **Problem Statement:** Show that in the finite complement topology on \( \mathbb{R} \), every subspace is compact. Is \( \mathbb{R} \) compact in the lower limit topology? **Concepts to Explore:** 1. **Finite Complement Topology:** - In this topology, the open sets are those whose complements in \( \mathbb{R} \) are finite, along with the empty set. 2. **Compactness:** - A subspace is compact if every open cover has a finite subcover. 3. **Lower Limit Topology:** - In this topology, open intervals are of the form \([a, b)\). **Discussion:** - **Compactness in Finite Complement Topology:** - Demonstrate why every subspace in this topology is compact by considering the nature of open sets and their complementarities. - **Compactness in Lower Limit Topology:** - Investigate whether \( \mathbb{R} \) with this topology is compact, considering the structure of the open intervals. Explore these concepts to deepen your understanding of topology and compactness properties within different topological spaces.
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