Sorgenfrey topology

right half-open interval topology

A topology $\tau$ on the real line $\mathbf R$ (cf. also Topological structure (topology)) defined by declaring that a set $G$ is open in $\tau$ if for any $x\in G$ there is an $\varepsilon_x>0$ such that $[x,x+\varepsilon_x)\subset G$. $\mathbf R$ endowed with the topology $\tau$ is termed the Sorgenfrey line, and is denoted by $\mathbf R^s$.

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 $X:=\mathbf R^s\times\mathbf R^s$ equipped with the product topology (cf. also Topological product), which is called the Sorgenfrey half-open square topology. Then $X$ 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 $\mathbf R$ 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 $f$ has a limit at the point $x$ with respect to the Sorgenfrey topology $\tau$ it has the same limit at $x$ with respect to the Euclidean topology when restricted to a $\tau$-neighbourhood of $x$). It has also the $G_\delta$-insertion property (given a subset $A$ of $\mathbf R$, there is a $G_\delta$-subset $G$ of $\mathbf R$ such that $G$ lies in between the $\tau$-interior and the $\tau$-closure of $A$). The Sorgenfrey topology satisfies the so-called essential radius condition: For any point $x$ and any $\tau$-neighbourhood $U_x$ of $x$ there is an "essential radius" $r(x,U_x)>0$ such that whenever the distance of two points $x$ and $y$ is majorized by $\min(r(x,U_x),r(y,U_y))$, then $U_x$ and $U_y$ intersect. The real line $\mathbf R$ equipped with the Sorgenfrey topology and the Euclidean topology is a binormal bitopological space, while $\mathbf R$ 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]	Steen, Lynn Arthur; Seebach, J.Arthur jun. Counterexamples in topology (2nd ed.) Springer (1978) ISBN 0-387-90312-7 Zbl 0386.54001