Dictionary order topology based query

Question

I am not sure i fully understand dictionary order topology...first picture still confusing me, why there is line in upward of a*b , why its jot downward.

and what will be diagram for (2*2,4*5).

Answer 1

Just apply the definition of the lexicographic order: the interval is all $x \times y$ with $a \times b < x \times y < c \times d$ (notation following Munkres, under protest..). To be bigger than $a \times b$ is to have $x$-coordinate $>a$ or $=a$ but then $y > b$. But we must also be smaller than $c \times d$ so $x < c$ or $x=c$ and $y < d$. So the vertical lines in between are all point where $a < x < c$. The points above $a \times b$ are the points with $x=a$ and $y > b$. The ones on the right are those with $x=c$ and $y < d$.

You can draw the similar picture for $2 \times 2$ and $4 \times 5$ yourself: vertical lines with $2<x<4$, some up-reaching part above $2 \times 2$ and so down-reaching part below $4 \times 5$. (the left point shifts downwards and the right one upwards; it's really the same).