Definition:Generalized Ordered Space/Definition 1

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Definition

Let $\struct {S, \preceq}$ be a totally ordered set.

Let $\tau$ be a topology on $S$.


$\struct {S, \preceq, \tau}$ is a generalized ordered space if and only if:

$(1): \quad \struct {S, \tau}$ is a Hausdorff space
$(2): \quad$ there exists a basis for $\struct {S, \tau}$ whose elements are convex in $S$.


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