Artificial intelligent assistant

Dictionary order topology based query 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). ![enter image description here](

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

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