Prove that a set is a topology. So I know the definition of topology but I find it really difficult to know what to do to prove a given set is a topology.
I was wondering if someone could talk me through a few problems.
$\tau_1=\{\emptyset, \Bbb R, (-a,a): a \in \Bbb R, a>0\}$
$\tau_2=\{\emptyset, \Bbb R, [-a,a]: a \in \Bbb R, a>0\}$
$\tau_3=\{\emptyset,\Bbb R,[−n,n],(−a,a) : a \in \Bbb R, a > 0, n\in \Bbb N>0\}$
(i) X and the empty set, Ø, belong to τ ,
(ii) the union of any (finite or infinite) number of sets in τ belongs to τ, and
(iii) the intersection of any two sets in τ belongs to τ.
Are the three things that X must satisfy to be a topology. 
Thanks
 A: Here $X=\mathbb R$. you should check this conditions(A is a subset of $P(\mathbb R))$:
1) $\emptyset \in A$
2)$\mathbb R \in A$
3)closed under finite intersections.
4)closed under arbitrary unions.
since, all the given sets satisfy 1,2, check the conditions 3,4.
A: So I'm sure you know but the following three axioms must hold for a set to be a topology:
(i) $\emptyset,X\in\tau$
(ii) $U_1,U_2\in\tau\Rightarrow U_1\cap U_2\in \tau$
(iii) $(U_\alpha)_{\alpha\in \Bbb A}\Rightarrow \cup_{\alpha\in\Bbb A}U_\alpha\in \tau$, where $\Bbb A$ is an arbitrary index.
So what we want to do is show that these axioms hold, if they don't then the collection is not a topology on $X$.
1st example: $X=\Bbb R$.
(i) does $\emptyset,\Bbb R\in \tau_1$? yes we can see from the set definition you have been given.
(ii) Let $U_1,U_2\in\tau_1$ where $U_1=(-b,b),U_2=(-a,a)$ with loss of generality we must have either $a\le b$ or $b\le a$, let us assume $a\le b$, then in fact $U_2\subset U_1$ 
Thus $U_1\cap U_2=U_2\in\tau_1$ thus (ii) holds.
(iii) Take the family $(U_\alpha)_{\alpha\in\Bbb A}$, where $U_\alpha=(-K_\alpha,K_\alpha)$ for some $K_\alpha\in\Bbb R$
We see that $\cup_{\alpha\in\Bbb A}U_\alpha=\cup_{\alpha\in\Bbb A}(-K_\alpha,K_\alpha)\in \tau_1$ since the arbitrary union of open intervals in $\Bbb R$ is an open interval. Thus (iii) holds
Thus $\tau_1$ is a topology on $\Bbb R$
