Sorgenfrey topology

right half-open interval topology

A topology on the real line (cf. also Topological structure (topology)) defined by declaring that a set is open in if for any there is an such that . endowed with the topology is termed the Sorgenfrey line, and is denoted by .

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). It is first countable but not second countable (cf. also First axiom of countability; Second axiom of countability). Moreover, the Sorgenfrey line is hereditarily Lindelöf, zero dimensional and paracompact (cf. also Lindelöf space; Zero-dimensional space; Paracompact space). Any compact subset of the Sorgenfrey line is countable and nowhere dense in the usual Euclidean topology (cf. Nowhere-dense set). The Sorgenfrey topology is neither locally compact nor locally connected (cf. also Locally compact space; Locally connected space).

Consider the Cartesian product equipped with the product topology (cf. also Topological product), which is called the Sorgenfrey half-open square topology. Then is completely regular but not normal (cf. Completely-regular space; Normal space). It is separable (cf. Separable space) but neither Lindelöf nor countably paracompact.

Many further properties of the Sorgenfrey topology are examined in detail in [a1]. Namely, the Sorgenfrey topology is a fine topology on the real line, and equipped with both the Sorgenfrey topology and the Euclidean topology serves as an example of a bitopological space (that is, a space endowed with two topological structures). The Sorgenfrey topology satisfies the condition (tFL) when studying fine limits (if a real-valued function has a limit at the point with respect to the Sorgenfrey topology it has the same limit at with respect to the Euclidean topology when restricted to a -neighbourhood of ). It has also the -insertion property (given a subset of , there is a -subset of such that lies in between the -interior and the -closure of ). The Sorgenfrey topology satisfies the so-called essential radius condition: For any point and any -neighbourhood of there is an "essential radius" such that whenever the distance of two points and is majorized by , then and intersect. The real line equipped with the Sorgenfrey topology and the Euclidean topology is a binormal bitopological space, while with the Sorgenfrey and the density topology is not binormal. See [a1] for answers to interesting questions concerning the class of continuous functions in the Sorgenfrey topology and for functions of the first or second Baire classes.

References

[a1]	J. Lukeš, J. Malý, L. Zajíček, "Fine topology methods in real analysis and potential theory" , Lecture Notes in Mathematics , 1189 , Springer (1986)
[a2]	R.H. Sorgenfrey, "On the topological product of paracompact spaces" Bull. Amer. Math. Soc. , 53 (1947) pp. 631–632
[a3]	A.S. Steen, J.A. Seebach Jr., "Counterexamples in topology" , Springer (1978)