Noll, Dominikus Topological spaces with a linear basis. (English) Zbl 0664.54019 Fundam. Math. 130, No. 2, 113-123 (1988). 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 Cited in 1 ReviewCited in 1 Document MSC: 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 54F65 Topological characterizations of particular spaces 54F15 Continua and generalizations Keywords:linear basis; connected orderable \(T_ 1\)-space Citations:Zbl 0272.54026 PDFBibTeX XMLCite \textit{D. Noll}, Fundam. Math. 130, No. 2, 113--123 (1988; Zbl 0664.54019) Full Text: DOI EuDML