Is every GO-topology a join of two orderable topologies?

Abstract:

As a generalization of ${\text {GO}}$-topologies (${\text {GO = generalized ordered}}$), we are interested in those topologies (here called ${\text {JO}}$-topologies) on a set $X$ which can be expressed as a join of orderable topologies (the join being taken in the lattice of all topologies on $X$). If a topology $t$ is the join of $m$ orderable topologies we say $t$ is ${\text {JO}}[m]$. It is not difficult to prove that every ${\text {GO}}$-topology is a ${\text {JO}}$-topology, but the question (raised in [M 71]) as to whether every ${\text {GO}}$-topology is ${\text {JO}}[2]$ seems much more difficult). We show that $X$ is ${\text {JO[2]}}$ if $X$ is a subspace of an orderable space $D$, where $D$ is either metrizable and locally separable, or connected with countable cellularity. (The theorem is actually more general than is stated here.) We give an example to show that for any positive integer $n$ there is a finite join of order topologies which is not ${\text {JO}}[n]$, but these are not ${\text {GO}}$-topologies.