Are family of subsets $\delta$ topology on $R$ number line?
$\delta$={${\emptyset},R,(-\infty,x),x\in R$}
$\delta$={${\emptyset},R,(-\infty,x),x\in Z$}
$\delta$={${\emptyset},R,(-\infty,x),x\in Q$}
In order to be topology it must fulfil $3$ criterion.
$1$)$\emptyset$ and $R$ must $\in$ $\delta$
$2$)Any number union of elements in $\delta$ must $\in$ $\delta$
$3$)Any finite number intersection of elements in $\delta$ must $\in$ $\delta$.
For every of $3$ examples first criterion holds.
For second criterion if we take $(-\infty,x)x\in R$ union will be some $(-\infty,x)$ where still $x\in R$
same for intersection and $Q$ and $Z$.
In my work 3 examples are topology.
Can you explain what is difference here because I think that I am doing something wrong here.