@MISC{TOPGEN_6.MIZ,
  AUTHOR = {St. Arnaud, Adam J.J. and Rudnicki, Piotr},
  TITLE = {Some Properties of the {S}orgenfrey Line
    and the {S}orgenfrey Plane},
  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.},
  DAY = {17},
  MONTH = {April},
  YEAR = {2013},
  ADDRESS1 = {Department of Computing Science\\University of Alberta\\116 St. and 85 Ave., Edmonton, AB, Canada},
  ADDRESS2 = {Department of Computing Science\\University of Alberta\\116 St. and 85 Ave., Edmonton, AB, Canada},
  SUMMARY = {We first provide a modified version of the proof
  in \cite{TOPGEN_5.ABS} that the Sorgenfrey line is $T_1$.
  Here, we proved that it is in fact $T_2$, a stronger result.
  This means that every pair of distinct points in the
  Sorgenfrey line can be separated by disjoint open neighbourhoods.
  Next, we proved that all subspaces of \verb!R^1!
  (that is the real line with the usual topology) are Lindelof, i.e.
  every open cover has a countable sub-cover. We utilize this result
  in the proof. This proof is based on the proof found in \cite{Engelking:1968}.
  Next, we construct the Sorgenfrey plane, as the product
  topology of the Sorgenfrey line and itself. We prove that the
  Sorgenfrey plane is NOT Lindelof, and therefore the product
  spaces need not be Lindelof. Further, we state that the Sorgenfrey
  line is regular, following from \cite{TOPGEN_5.ABS}:59,
  since Tychonoff implies regular. Then we state that the
  Sorgenfrey line is normal since it is both regular and
  Lindelof. Normal spaces ensure
  that every pair of disjoint closed sets can be separated by
  disjoint open neighbourhoods. 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 normal and many of the lemmas
  leading up to this result, are modelled after the
  proof in \cite{TOPGEN_5.ABS}, that the Niemytzki-plane
  is not normal. Information was also gathered from
  \cite{SteenSeebach:1978}.}}

@BOOK{Engelking:1968,
 AUTHOR={Engelking, Ryszard},
 TITLE={Outline of General Topology},
 PUBLISHER={North-Holland Publishing Company},
 YEAR=1968}

@BOOK{SteenSeebach:1978,
 AUTHOR={Steen, Lynn Arthur and Seebach, J. Arthur Jr.},
 TITLE={Counterexamples in Topology},
 PUBLISHER={Springer-Verlag},
 YEAR=1978}
