Struggling to conceptually think about lexicographically ordered square

Question

I have been given the task to determine the closure of several subsets of the lexicographically ordered square, and am having an immense amount of trouble picturing the open sets. I think about an open interval containing a point with $y=1$ in the ordered pair as going up and "hitting" the top of the square before sort of sprouting back up from the closest point nearby from the $x$-axis. Almost like a line meeting the top, and then a small line coming up, right near that line. My question is, is it wrong to think about it this way, must an open interval on this square contain at least infinitely many vertical lines packed next to the "starting line" for the interval. If anyone could lead me to a sort of picture of open sets in this square, I would be more than happy.

Answer 1

Yes, it’s wrong, because there is nothing even remotely resembling a ‘next’ vertical segment.

Let $X$ be the lexicographically ordered square, let $\preceq$ be the lexicographic order, and suppose that $\langle a,1\rangle\in X$, where $0\le a<1$. If $p=\langle x_0,y_0\rangle,q=\langle x_1,y_1\rangle\in X$, and $(p,q)_\preceq$ is an open interval in $X$ containing $\langle a,1\rangle$, then

$$p=\langle x_0,y_0\rangle\prec\langle a,1\rangle\prec\langle x_1,y_1\rangle=q\,.$$

Since $\langle a,1\rangle\prec\langle x_1,y_1\rangle$, we know that either $a=x_1$ and $1<y_1$, or $a<x_1$, and the first of these is impossible, so $a<x_1$. This means that if $\langle u,v\rangle\in X$ and $a<u<x_1$, then

$$\langle a,1\rangle\prec\langle u,v\rangle\prec\langle x_1,0\rangle\preceq\langle x_1,y_1\rangle\,:$$

the open interval $(p,q)_\preceq$ contains the entire strip $(a,x_1)\times[0,1]$ lying between $x=a$ and $x=x_1$. This isn’t just infinitely many vertical lines: it’s all of the vertical lines in a whole ordinary open interval of the $[0,1]$.

The open interval $(p,q)_\preceq$ in $X$ also contains at least a half open interval at the top of the segment $\{a\}\times[0,1]$, a set of the form $\{a\}\times(b,q]$ for some $b\in[0,1)$. Specifically, if $p=\langle a,y_0\rangle$ and $y_1>0$, then

$$(p,q)_\preceq=\big(\{a\}\times(y_0,1]\big)\cup\big((a,x_1)\times[0,1]\big)\cup\big(\{x_1\}\times[0,y_1)\big)\,.$$

(If $y_1=0$, simply omit $\{x_1\}\times[0,y_1)$, as it is then empty.)

The diagram in this answer shows an open interval in $X$ around the point $\left\langle\frac12,0\right\rangle$; what happens at a point $\langle a,1\rangle$ is similar, except that the open interval has to extend to the right rather than to the left. If you turn that picture upside down, you get an open interval around $\left\langle\frac12,1\right\rangle$.

If $x_0<a$, the open interval $(p,q)_\preceq$ will also contain a vertical strip to the left of $x=a$ and possibly an open interval at the top end of a vertical segment at the left side of that strip.