What collection of intervals could not be a topology?

Question

I got these sets and for me, all of them are topologies of $R$, but I'm confused about this exercise, I think it should have sets that are not a topology. Each $\tau_i$ consists of $R$ and $\emptyset$ and every interval: $$a)\, (-r,\,r)\,\text{r is positive rational number}\,$$ $$b)\, [-r,\,r]\,\text{r is positive rational number }\,$$ $$c)\, (-r,\,r)\,\text{r is positive irrational number}$$ $$d)\, [-r,\,r]\,\text{r is positive irrational number}$$ $$e)\, [-r,\,r)\,\text{r is positive real number}\,\,\,\,\,\,\,\,\,\,\,\,$$ $$f)\, (-r,\,r]\,\text{r is positive real number}\,\,\,\,\,\,\,\,\,\,\,\,$$ $$g)\, [-r,\,r]\, \text{ and } (-r,r) \,\text{r is positive real}\,\,\,\,\,\,$$ $$h)\, [-n,\,n]\, \text{ and } (-r,r) \,\text{n is positive natural, r is positive real}$$ Every $\tau_i$ satisfies axiom $i)$, the intersection of two any intervals from above (for each case) is clearly one of the two intervals being intersected so belongs to $\tau$, the finite union is the same argument and the infinite union is $R$ so they satisfy the axioms but the exercise asks for those who are topologies, is there any of them that is not a topology?

Answer 1

HINT: Let $1=r_1<r_2<r_3<\ldots$, where $r_n$ are rational and $\lim_n r_n=\sqrt{2}$. Does $$ \bigcup_{n=1}^{\infty}(-r_n,r_n) $$ belong to the set in a)?

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