order topology and discrete topology

Question

I have this homework question. Consider the set $X = \{1,2,3\}$.

(a) With the natural order on $X$, find the basis for its order topology, (b) Show that the order topology on $X$ equals its discrete topology.

I suppose the natural order to be $1<2<3$, so that $1$ the is the least element and $3$ is the largest element, then $B=\{[1,3),(1,3),(1,3]\}$ is the basis for the order topology on $X$.

For part (b), I would like to write $B=\bigg\{\{1,2\},\{2\},\{2,3\}\bigg\}$ but I see it will not satisfy. I need help! Thanks.

Answer 1

I hope we are working with the same definition of order topology.

According to this defintion, the base contains all intervals $(a,b)$, $(a,\infty)$, $(-\infty,b)$. (Of course, the base for a topological space is not determinecd uniquely, but this is the one from definition.) Since this set has largest and smallest element, you can rewrite them as (a,b), (a,3], [1,b).

Since, [1,2)={1}, (1,3)={2}, (2,3]={3}, the base contains all singletons and thus the space is discrete.