What is the topology generated by \EuScript{E} for X = \mathbb{R}?

[complexnumber]

Homework Statement

Let (X,\tau) be X = \mathbb{R} equipped with the topology
generated by \EuScript{E} := \{[a,\infty) | a \in \mathbb{R} \}.

Show that \tau = \{ \varnothing, \mathbb{R} \} \cup \{<br /> [a,\infty), (a, \infty) | a \in \mathbb{R} \}

Homework Equations

A topology generated by \EuScript{E} is \tau(\EuScript{E}) = \bigcap \{ \tau \subset \mathcal{P}(X) | \tau \text{ is a topology } \wedge \tau \supset \EuScript{E} \}

The Attempt at a Solution

I can see that \tau = \{ \varnothing, \mathbb{R} \} \cup \{<br /> [a,\infty), (a, \infty) | a \in \mathbb{R} \} is a topology for X. But I don't know why the generated topology contains (a,\infty) as well. How is this obtained? How should I prove that \tau = \{ \varnothing, \mathbb{R} \} \cup \{<br /> [a,\infty), (a, \infty) | a \in \mathbb{R} \} is the intersection of all topologies containing \EuScript{E}?

 

[jbunniii]

But I don't know why the generated topology contains (a,\infty) as well. How is this obtained?

Consider the sets

\left[a + \frac{1}{n}, \infty\right)

for n = 1,2,\ldots

What is the union of these sets?

 

[complexnumber]

Consider the sets

\left[a + \frac{1}{n}, \infty\right)

for n = 1,2,\ldots

What is the union of these sets?

I see. The union of these sets is (a,\infty). Hence (a,\infty) must be in the topology in order to satisfy the closed under arbitrary union condition.

Thanks very much for your help.