### Exploring Lower Limit Topology vs Standard Topology on ℝIn the realm of topology, various topologies can be imposed on the set of real numbers, ℝ. In this section, we will investigate properties of the lower limit topology, specifically how it compares to the standard topology on ℝ.We aim to demonstrate that the lower limit topology on ℝ is strictly finer than the standard topology. This means that the lower limit topology contains more open sets than the standard topology, effectively making it a more "detailed" or "refined" structure. We will break this into two parts for clarity:1. **Show that all sets that are open in the standard topology are open in the lower limit topology:** - The standard topology on ℝ is typically defined using open intervals (a, b), where any point within this interval has a neighborhood entirely contained within (a, b). In contrast, the lower limit topology uses half-open intervals of the form [a, b), including the left endpoint but excluding the right. - Since any open interval (a, b) in the standard topology can be expressed as a union of half-open intervals [a+ε, b), where ε > 0 is small enough, every open set in the standard topology can be represented as a union of open sets in the lower limit topology. Hence, each open set in the standard topology is also open in the lower limit topology.2. **Provide an example of a set that is open in the lower limit topology but not in the standard topology:** - Consider a half-open interval of the form [a, b) within ℝ. In the lower limit topology, such an interval [a, b) is open by definition. - However, in the context of the standard topology, the set [a, b) is not open, because any neighborhood around the point a must include points less than a, which aren't contained in [a, b).### Example:- **Set Example in Lower Limit Topology**: The interval [0,1) is open in the lower limit topology.- **Justification**: In the lower limit topology, [0,1) is an open set by definition.- **Standard Topology Perspective**: In the standard topology, any open set containing 0 must also include points of the form (-ε, 0) for some small ε > 0. Since these points are not in

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Show that the lower limit topology on R is strictly finer than the standard topology on R. That is: • Show that all sets that are open in the standard topology are open in the lower limit topology and • Also write an example of a set that is open in the lower limit topology that is not open in the standard topology. Justify why this example set is not open in the standard topology.

Show that the lower limit topology on R is strictly finer than the standard topology on R. That is: • Show that all sets that are open in the standard topology are open in the lower limit topology and • Also write an example of a set that is open in the lower limit topology that is not open in the standard topology. Justify why this example set is not open in the standard topology.

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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