<abstract xmlns="http://www.w3.org/1999/xhtml">We first provide a modified version of the proof in [3] that
the Sorgenfrey line is <italic>T</italic>1. 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].</abstract>

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].

Some Properties of the Sorgenfrey Line and the Sorgenfrey Plane

Formalized Mathematics

{"article-title":"Some Properties of the Sorgenfrey Line and the Sorgenfrey Plane"}

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...

Some Properties of the Sorgenfrey Line and the Sorgenfrey Plane

Pubblicato online: 01 giu 2013

Pagine: 83 - 85

DOI: https://doi.org/10.2478/forma-2013-0009

Parole chiave
topological spaces, products of normal spaces, Sorgenfrey line, Sorgenfrey plane, Lindelöf spaces

This content is open access.

Some Properties of the Sorgenfrey Line and the Sorgenfrey Plane

Pubblicato online: 01 giu 2013

Pagine: 83 - 85

DOI: https://doi.org/10.2478/forma-2013-0009

Parole chiavetopological spaces, products of normal spaces, Sorgenfrey line, Sorgenfrey plane, Lindelöf spaces

This content is open access.

Parole chiave
topological spaces, products of normal spaces, Sorgenfrey line, Sorgenfrey plane, Lindelöf spaces