A closed form for a particular topology. I am trying to find some sort of 'closed form' (if possible) of a particular topology generated by the sets: $({x\in\mathbb{R}\ \vert x\geq a}), a\in \mathbb {R}$. 
Thanks !
 A: Note that given $a \in \mathbb{R}$ we have $\{ x \in \mathbb{R} : x \geq a \} = [a,+\infty)$, and so your family $\mathcal{B}$ consists of all "closed rays to $+\infty$."
Next, $\mathcal{B}$ is a base for a topology on $\mathbb{R}$:


*

*Clearly $a \in [ a , + \infty ) \in \mathcal{B}$ for each $a \in \mathbb{R}$; and

*Given $a , b \in \mathbb{N}$ we have that $[a,+\infty) \cap [b,+\infty) = [ \max \{a,b\} , + \infty ) \in \mathcal{B}$ for $a,b \in \mathbb{R}$,


Therefore the topology generated consists of all unions of elements of $\mathcal{B}$.
At this point it is fairly easy to conclude that this topology consists of all "rays to $+\infty$"; more exactly, the topology is
$$\mathcal{T} = \{ \varnothing, \mathbb{R} \}  \cup \{ [ a , + \infty ) : a \in \mathbb{R} \} \cup \{ ( a , + \infty ) : a \in \mathbb{R} \}$$
Clearly the set $\varnothing$ and $\mathbb{R}$ must be open, and your basis consists of all sets of the form $[ a , +\infty )$. As $( a , + \infty ) = \bigcup_{n \geq 1} [ a + \frac 1n , + \infty )$, these sets are also open.  
If $U \subseteq \mathbb{R}$ is nonempty, open, and $a \in U$ it follows that $[ a , + \infty ) \subseteq U$.  If $\inf U = - \infty$ it follows that $U = \mathbb{R}$.  Otherwise, letting $a_0 = \inf ( U )$ we have that either $a_0 \in U$ (whence $U = [ a_0 , +\infty )$) or $a_0 \notin U$ (whence $U = ( a_0 , +\infty )$). 
