Some Properties of the Sorgenfrey Line and the Sorgenfrey Plane
We first provide a modified version of the proof in  that the Sorgenfrey line is T1. Here, we prove that it is in fact T2, a stronger result. Next, we prove that all subspaces of ℝ1 (that is the real line with the usual topology) are Lindel¨of. We utilize this result in the proof that the Sorgenfrey line is Lindel¨of, which is based on the proof found in . Next, we construct the Sorgenfrey plane, as the product topology of the Sorgenfrey line and itself. We prove that the Sorgenfrey plane is not Lindel¨of, and therefore the product space of two Lindel¨of spaces need not be Lindel¨of. Further, we note that the Sorgenfrey line is regular, following from :59. Next, we observe that the Sorgenfrey line is normal since it is both regular and Lindel¨of. Finally, we prove that the Sorgenfrey plane is not normal, and hence the product of two normal spaces need not be normal. The proof that the Sorgenfrey plane is not normal and many of the lemmas leading up to this result are modelled after the proof in , that the Niemytzki plane is not normal. Information was also gathered from .
I would like to thank Piotr Rudnicki for taking me on as his summer student and being a mentor to me. Piotr was an incredibly caring, intelligent, funny, passionate human being. I am proud to know I was his last student, in a long line of students he has mentored and cared about throughout his life. Thank you Piotr, for the opportunity you gave me, and for the faith, confidence and trust you showed in me. I will miss you.
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