**The Sorgenfrey Line and Lower Limit Topology****Basis for Topology on $\mathbb{R}$:** Consider the set $\{ [a, b): \ a < b \}$ as a basis for a topology on the real number line $\mathbb{R}$. **Lower Limit Topology:** The resulting topology from this basis is known as the lower limit topology. In this topology, each basic open set is a half-open interval $[a, b)$, where $a < b$.**Sorgenfrey Line:** The real line endowed with the lower limit topology is referred to as the Sorgenfrey line, commonly denoted by $\mathbb{R}_l$.Understanding topological bases such as this allows for deeper insight into different topological structures and their properties. The Sorgenfrey line, $\mathbb{R}_l$, is an essential example in topology, illustrating the utility and uniqueness of different bases for topologies.

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Show that {|a, b): a < b}is a basis for a topology on R.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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**The Sorgenfrey Line and Lower Limit Topology**

**Basis for Topology on $\mathbb{R}$:** Consider the set $\{ [a, b): \ a < b \}$ as a basis for a topology on the real number line $\mathbb{R}$.

**Lower Limit Topology:** The resulting topology from this basis is known as the lower limit topology. In this topology, each basic open set is a half-open interval $[a, b)$, where $a < b$.

**Sorgenfrey Line:** The real line endowed with the lower limit topology is referred to as the Sorgenfrey line, commonly denoted by $\mathbb{R}_l$.

Understanding topological bases such as this allows for deeper insight into different topological structures and their properties. The Sorgenfrey line, $\mathbb{R}_l$, is an essential example in topology, illustrating the utility and uniqueness of different bases for topologies.

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