Cartesian product of linear continua

Question

Can product of two linear continua with order topology be a linear continuum?

Clearly it satisfies the denseness property.

So question is equivalent to proving whether it is connected or not.

Answer 1

If we give $X \times Y$ (where $X,Y$ are linear continua) the lexicographic product order (so $(x,y) < (x',y')$ iff $x < x'$ or ($x=x'$ and $y < y'$)) the result need not be connected: take $X = Y = \mathbb{R}$ in the usual order. The resulting lexicographically ordered plane is not a linear continuum (not connected) as e.g. $\{0\} \times \mathbb{R}$ has an upper bound (e.g $(1,0)$) but no supremum, i.e. smallest upper bound (if $(x,y)$ is an upper bound, $(x,y-1)$ is a smaller one). In the case of $[0,1] \times [0,1]$ in the lexicographic order, the result is connected, and a linear continuum.

In fact this ($X \times Y, <_{\text{lex}})$ is a continuum) will hold iff $Y$ has both a minimum and a maximum.

The lexicographic product can be a linear continuum even when $X$ is not, e.g. take $X = \omega_1$ and $Y = [0,1)$ which has as its "product" the long line, a linear continuum. Or take $X = \mathbb{N}$ and we get $[0,\infty)$, also a linear continuum.