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Locally ordered topological spaces

Pikul Piotr

Reports on Mathematical Logic

2020

Vol. 55

113--141

While topology given by a linear order has been extensively studied, this cannot be said about the case when the order is given only locally. The aim of this paper is to fill this gap. We consider relation between local orderability and separation axioms and give characterisation of those regularly locally ordered spaces which are connected, locally connected or Lindel¨of. We prove that local orderability is hereditary on open, connected or compact subsets. A collection of interesting examples is also offered.

A covering criterion and its applications to product theorems

Jachymski J.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2005

Vol. 45, [Z] 1

35--43

We establish a covering criterion involving a neighbourhood system and idealsof open sets which yields,in particular, a compactness criterion for an arbitrary topologicalspace. As an application, we give new proofsof Tychonoff's compactness theorem: we consider separately the case of acountable product, in a proof of which the ordinary mathematical induction isused, and the case of an uncountable product proved by the transfiniteinduction. Subsequently, the same argument is applied to obtain some resultson products of Lindelöf spaces.