Linearly ordered extensions of GO spaces

AbstractWe prove a main theorem: Theorem. There always exists a minimal linearly ordered d-extension of a GO space, where a LOTS Y is said to be a linearly ordered d-extension of a GO space 〈 X, τ, ≤〉 if Y contains X as a dense subspace and the ordering of Y extends the ordering ≤ of X. As some applications of the Theorem, (1) we give a partial negative answer to a problem: “Does every perfect GO space have a perfect orderable d-extension?” (2) For a discrete space 〈X, τ〉 of cardinality ω1, there is a linear ordering ≤ of X such that 〈X, τ, ≤〉 is a GO space and whose every linearly ordered d-extension contains an order preserving copy of the ordinal space ω1 as a dense subspace