Sorgenfrey line

Definition

The Sorgenfrey line is defined as follows: as a set, it is the real line, and its basis of open sets is taken as all the half-open, half-closed sets, viz sets of the form $[a,b)$. Equivalently, we can say that it is obtained by giving the lower limit topology corresponding to the usual ordering on the real line.

Topological space properties

Properties it does satisfy

• Totally disconnected space: The Sorgenfrey line is totally disconnected; given any two points, we can separate them by disjoint open sets.
• First-countable space
• Separable space
• Lindelof space
• Paracompact space
• Baire space
• Perfectly normal space

Properties it does not satisfy

• Sigma-compact space
• Locally compact space
• Metrizable space