Q:

Why the order square is not path-connected

In the book of Topology by Munkres, at page 156, it is given and proven that 

The ordered square $I_0^2$ (no idea why it is shown as the square of
  $I_0$) is connected but not path connected.

I understood the given proof, but cannot imagine which pair of point might causing the issue that this space is not path connected, so can you point me two elements in this space where there is no path connecting these two points ?

A:

Intuitively, if two points have different $x$ coordinates then in order to construct a path that respects the lexicographic ordering you would need a path that either passes through the top of the square or bottom of the square and join up with the "next" number on the $x$-axis somehow. Since the concept of "next" number in the reals is nonsense so to is the possibility of path connecting the space.

