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Topological spaces with a linear basis. (English) Zbl 0664.54019

A linear basis for a topological space is a basis \({\mathcal B}\) such that (1) the union of two intersecting elements of \({\mathcal B}\) is an element of \({\mathcal B}\) and (2) if \(\{\) A,B,C\(\}\) is a simple chain of elements of \({\mathcal B}\) with end links A and C, then each element of \({\mathcal B}\) that intersects B and is not a subset of B intersects either A or C. A topological space is said to be orderable if its topology is induced by a liner order relation. It is easily seen that an orderable space or a (generalized) simple closed curve has a linear basis. The main result of this paper is the following elegant theorem. If a connected \(T_ 2\)- space has a linear basis, then it is either orderable or a simple closed curve. Several known theorems are obtained as corollaries to the main theorem, including a theorem of J. van Dalen and E. Wattel [General Topol. Appl. 3, 347-354 (1973; Zbl 0272.54026)] characterizing a connected orderable \(T_ 1\)-space in terms of a subbasis. Two examples are given to show that the notion of linear basis is inappropriate for the characterization of nonconnected orderable spaces.
Reviewer: B.J.Pearson

MSC:

54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54F65 Topological characterizations of particular spaces
54F15 Continua and generalizations

Citations:

Zbl 0272.54026
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