# Show $\{\emptyset,\{0,1,2\},\{0\},\{1,2\}\}$ a topology on the set $\{0,1,2\}$?

I have a set $$X=\{0,1,2\}$$ and I know that $$\tau=\{\emptyset,X,\{0\},\{1,2\}\}$$ is a topology on the set X.

I know the axioms required to show it is a topology.

$\emptyset$ and X are both in $\tau$ so the first axiom is satisfied. This is clear.

The second axiom says that the intersection of any finite members of $\tau$ is in $\tau$. The third axiom says that the union of any collection of elements of $\tau$ is in $\tau$.

I can not clearly see how the second and third properties come into play, so how do I directly find the unions and intersections involved with the given example?

• Just directly verify them. It's not hard to find the intersections and unions involved. Oct 3, 2015 at 10:56
• What is $\phi$? If you mean the empty set, that's $\emptyset$, obtained by \emptyset. Oct 3, 2015 at 11:01
• @celtschk thanks for the tip Oct 3, 2015 at 11:03
• The line that says "$\phi$ and $\tau$ are both in $\tau$ should be "$\emptyset$ and $X$ are both in $\tau$"
– R_D
Oct 3, 2015 at 11:03
• Maybe it helps you to notice that since $\tau$ is a finite sets, any intersections of its members are finite intersections. Same for unions. Oct 3, 2015 at 11:08

It suffices to construct unions and intersections of all members of $\tau$ explicitly. Since the unions and intersections of any element with $\emptyset$ and $X$ are trivial, it suffices to see that $\left\{ 0 \right\} \cap \left\{ 1,2 \right\} = \emptyset$ and $\left\{ 0 \right\} \cup \left\{ 1,2 \right\}= X$, which both lie in $\tau$.