Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
Online
ISSN
1898-9934
See all formats and pricing
More options …
Volume 21, Issue 2 (Jun 2013)

Issues

Some Properties of the Sorgenfrey Line and the Sorgenfrey Plane

Adam St. Arnaud
  • University of Alberta Edmonton, Canada
/ Piotr Rudnicki
  • University of Alberta Edmonton, Canada

Summary

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].

Acknowledgement:

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.

Keywords: topological spaces; products of normal spaces; Sorgenfrey line; Sorgenfrey plane; Lindelöf spaces

  • [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Google Scholar

  • [2] Grzegorz Bancerek. On constructing topological spaces and Sorgenfrey line. Formalized Mathematics, 13(1):171-179, 2005.Google Scholar

  • [3] Grzegorz Bancerek. Niemytzki plane - an example of Tychonoff space which is not T4. Formalized Mathematics, 13(4):515-524, 2005.Google Scholar

  • [4] Grzegorz Bancerek. Bases and refinements of topologies. Formalized Mathematics, 7(1): 35-43, 1998.Google Scholar

  • [5] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Google Scholar

  • [6] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Google Scholar

  • [7] Agata Darmochwał and Yatsuka Nakamura. Metric spaces as topological spaces - fundamental concepts. Formalized Mathematics, 2(4):605-608, 1991.Google Scholar

  • [8] Ryszard Engelking. Outline of General Topology. North-Holland Publishing Company, 1968.Google Scholar

  • [9] Adam Grabowski. On the boundary and derivative of a set. Formalized Mathematics, 13 (1):139-146, 2005.Google Scholar

  • [10] Adam Grabowski. On the Borel families of subsets of topological spaces. Formalized Mathematics, 13(4):453-461, 2005.Google Scholar

  • [11] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Google Scholar

  • [12] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Google Scholar

  • [13] Karol Pak. Basic properties of metrizable topological spaces. Formalized Mathematics, 17(3):201-205, 2009. doi:10.2478/v10037-009-0024-8.CrossrefGoogle Scholar

  • [14] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Google Scholar

  • [15] Lynn Arthur Steen and J. Arthur Jr. Seebach. Counterexamples in Topology. Springer- Verlag, 1978.Google Scholar

  • [16] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4): 535-545, 1991.Google Scholar

  • [17] Andrzej Trybulec. Subsets of complex numbers. Mizar Mathematical Library.Google Scholar

  • [18] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Google Scholar

  • [19] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Google Scholar

  • [20] Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231-237, 1990.Google Scholar

About the article

Published in Print: 2013-06-01


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/forma-2013-0009.

Export Citation

This content is open access.

Comments (0)

Please log in or register to comment.
Log in