# Is the Sorgenfrey line separable?

## Is the Sorgenfrey line separable?

This topological space is known as the Sorgenfrey line. The rational numbers are a dense subset of this topological space, so the Sorgenfrey line is separable.

## Is Sorgenfrey a compact line?

We observe also that the Sorgenfrey line cannot be compact (since the usual topology on R is coarser and not compact). However, the Sorgenfrey line is hereditarily Lindelöf, i.e. every subspace is Lindelöf (Exercise).

Is the Sorgenfrey line Lindelof?

The Sorgenfrey line E is Lindelöf. Proof. Let c be a basic open (in E) cover of R. Since R with the usual topology is second-countable, it is Lindelöf.

### Is R with lower limit topology locally compact?

with the lower limit topology is not locally compact.

### Is the Sorgenfrey line a subspace of a separable space?

This has a proof that for any ordered space separable implies hereditarily separable. And the Sorgenfrey line is a subspace of a separable ordered space (e.g. the double arrow). Thanks for contributing an answer to Mathematics Stack Exchange!

Which is an example of the Sorgenfrey line?

The Sorgenfrey line serves as a counterexample to several topological properties, see, for example, [a3]. For example, it is not metrizable (cf. also Metrizable space) but it is Hausdorff and perfectly normal (cf. also Hausdorff space; Perfectly-normal space ).

## Is the Sorgenfrey line a Lindelof space?

The product of two Lindelöf spaces need not be Lindelöf. For example, the Sorgenfrey line is not Lindelöf. In a Lindelöf space, every locally finite family of nonempty subsets is at most countable. A space is hereditarily Lindelöf if and only if every open subspace of it is Lindelöf.

## Is the Sorgenfrey half open square topology locally compact?

The Sorgenfrey topology is neither locally compact nor locally connected (cf. also Locally compact space; Locally connected space ). Consider the Cartesian product $X:=\\mathbf R^s imes\\mathbf R^s$ equipped with the product topology , which is called the Sorgenfrey half-open square topology.