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Some properties of the Sorgenfrey line and the Sorgenfrey plane. (English) Zbl 1298.54013

Summary: We first provide a modified version of the proof in [G. Bancerek, “Niemytzki plane – an example of Tychonoff space which is not \(T_4\)”, Formaliz. Math. 13, No. 4, 515–524 (2005)] that the Sorgenfrey line is \(T_{1}\). Here, we prove that it is in fact \(T_2\), a stronger result. Next, we prove that all subspaces of \(\mathbb{R}^{1}\) (that is the real line with the usual topology) are Lindelöf. We utilize this result in the proof that the Sorgenfrey line is Lindelöf, which is based on the proof found in [R. Engelking, Outline of general topology. Amsterdam: North-Holland (1968; Zbl 0157.53001)]. Next, we construct the Sorgenfrey plane, as the product topology of the Sorgenfrey line and itself. We prove that the Sorgenfrey plane is not Lindelöf, and therefore the product space of two Lindelöf spaces need not be Lindelöf. Further, we note that the Sorgenfrey line is regular, following from [Bancerek, loc. cit.]. Next, we observe that the Sorgenfrey line is normal since it is both regular and Lindelöf. 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 [Bancerek, loc. cit.], that the Niemytzki plane is not normal. Information was also gathered from [L. A. Steen and J. A. Seebach jun., Counterexamples in topology. New York, Heidelberg, Berlin: Springer (1978; Zbl 0386.54001)].

MSC:

54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54B10 Product spaces in general topology
03B35 Mechanization of proofs and logical operations
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