We first provide a modified version of the proof in [3] 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 [8]. 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 [3]: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 [3], that the Niemytzki plane is not normal. Information was also gathered from [15].
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.
@article{bwmeta1.element.doi-10_2478_forma-2013-0009,
author = {Adam St. Arnaud and Piotr Rudnicki},
title = {Some Properties of the Sorgenfrey Line and the Sorgenfrey Plane},
journal = {Formalized Mathematics},
volume = {21},
year = {2013},
pages = {83-85},
zbl = {1298.54013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_forma-2013-0009}
}
Adam St. Arnaud; Piotr Rudnicki. Some Properties of the Sorgenfrey Line and the Sorgenfrey Plane. Formalized Mathematics, Tome 21 (2013) pp. 83-85. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_forma-2013-0009/
[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
[2] Grzegorz Bancerek. On constructing topological spaces and Sorgenfrey line. Formalized Mathematics, 13(1):171-179, 2005.
[3] Grzegorz Bancerek. Niemytzki plane - an example of Tychonoff space which is not T4. Formalized Mathematics, 13(4):515-524, 2005.
[4] Grzegorz Bancerek. Bases and refinements of topologies. Formalized Mathematics, 7(1): 35-43, 1998.
[5] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
[6] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
[7] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.
[8] Ryszard Engelking. Outline of General Topology. North-Holland Publishing Company, 1968.
[9] Adam Grabowski. On the boundary and derivative of a set. Formalized Mathematics, 13 (1):139-146, 2005.
[10] Adam Grabowski. On the Borel families of subsets of topological spaces. Formalized Mathematics, 13(4):453-461, 2005.
[11] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.
[12] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
[13] Karol Pak. Basic properties of metrizable topological spaces. Formalized Mathematics, 17(3):201-205, 2009. doi:10.2478/v10037-009-0024-8.[Crossref]
[14] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.
[15] Lynn Arthur Steen and J. Arthur Jr. Seebach. Counterexamples in Topology. Springer- Verlag, 1978. | Zbl 0386.54001
[16] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4): 535-545, 1991.
[17] Andrzej Trybulec. Subsets of complex numbers. Mizar Mathematical Library.
[18] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
[19] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
[20] Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231-237, 1990.