Another approach for lexicographic GO-spaces

It is known that for a GO-space X, there is the smallest LOTS $X^{⁎}$ containing X as a dense subspace, that is, if a LOTS L contains X as a dense subspace, then L contains $X^{⁎}$.

Also lexicographic products of LOTS', which is called lexicographic LOTS', have been well-discussed. Recently, the notion of lexicographic products of GO-spaces was defined as follows:

for a sequence $\{X_{\alpha }:\alpha <\gamma \}$ of GO-spaces, the lexicographic GO-space $X={\prod }_{\alpha <\gamma }{X}_{\alpha }$ means the subspace X of the lexicographic LOTS $\hat{X}={\prod }_{\alpha <\gamma }{X}_{\alpha }^{⁎}$.

It is known that for a GO-space X, there is a well-known LOTS $X^{\diamond }$ containing X as a closed subspace. In this paper, first we show the LOTS $X^{\diamond }$ has the following nice property:

• if a LOTS L contains X as a closed subspace, then L contains $X^{\diamond }$.

Using this property, it is natural to define another notion of lexicographic GO-spaces as follows:

for a sequence $\{X_{\alpha }:\alpha <\gamma \}$ of GO-spaces, the lexicographic GO-space $X={\prod }_{\alpha <\gamma }{X}_{\alpha }$ means the subspace X of the lexicographic LOTS $\check{X}={\prod }_{\alpha <\gamma }{X}_{\alpha }^{\diamond }$.

We will see:

• the GO-space X as a subspace of $\hat{X}$ coincides with the GO-space X as a subspace of $\check{X}$,

• $X^{\diamond }$ is contained in $\check{X}$.

• we characterize that the lexicographic GO-space X is closed in $\check{X}$.