View source for Sorgenfrey topology

''right half-open interval topology''

A topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s1304301.png" /> on the real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s1304302.png" /> (cf. also [[Topological structure (topology)|Topological structure (topology)]]) defined by declaring that a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s1304303.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s1304304.png" /> if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s1304305.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s1304306.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s1304307.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s1304308.png" /> endowed with the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s1304309.png" /> is termed the Sorgenfrey line, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043010.png" />.

The Sorgenfrey line serves as a counterexample to several topological properties, see, for example, [[#References|[a3]]]. For example, it is not metrizable (cf. also [[Metrizable space|Metrizable space]]) but it is Hausdorff and perfectly normal (cf. also [[Hausdorff space|Hausdorff space]]; [[Perfectly-normal space|Perfectly-normal space]]). It is first countable but not second countable (cf. also [[First axiom of countability|First axiom of countability]]; [[Second axiom of countability|Second axiom of countability]]). Moreover, the Sorgenfrey line is hereditarily Lindelöf, zero dimensional and paracompact (cf. also [[Lindelöf space|Lindelöf space]]; [[Zero-dimensional space|Zero-dimensional space]]; [[Paracompact space|Paracompact space]]). Any compact subset of the Sorgenfrey line is countable and nowhere dense in the usual Euclidean topology (cf. [[Nowhere-dense set|Nowhere-dense set]]). The Sorgenfrey topology is neither locally compact nor locally connected (cf. also [[Locally compact space|Locally compact space]]; [[Locally connected space|Locally connected space]]).

Consider the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043011.png" /> equipped with the product topology (cf. also [[Topological product|Topological product]]), which is called the Sorgenfrey half-open square topology. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043012.png" /> is completely regular but not normal (cf. [[Completely-regular space|Completely-regular space]]; [[Normal space|Normal space]]). It is separable (cf. [[Separable space|Separable space]]) but neither Lindelöf nor countably paracompact.

Many further properties of the Sorgenfrey topology are examined in detail in [[#References|[a1]]]. Namely, the Sorgenfrey topology is a [[Fine topology|fine topology]] on the real line, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043013.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043014.png" /> has a limit at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043015.png" /> with respect to the Sorgenfrey topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043016.png" /> it has the same limit at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043017.png" /> with respect to the Euclidean topology when restricted to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043018.png" />-neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043019.png" />). It has also the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043021.png" />-insertion property (given a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043023.png" />, there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043024.png" />-subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043027.png" /> lies in between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043028.png" />-interior and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043029.png" />-closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043030.png" />). The Sorgenfrey topology satisfies the so-called essential radius condition: For any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043031.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043032.png" />-neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043034.png" /> there is an  "essential radius"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043035.png" /> such that whenever the distance of two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043037.png" /> is majorized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043038.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043040.png" /> intersect. The real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043041.png" /> equipped with the Sorgenfrey topology and the Euclidean topology is a binormal bitopological space, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130430/s13043042.png" /> with the Sorgenfrey and the [[Density topology|density topology]] is not binormal. See [[#References|[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|Baire classes]].

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