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 $$\tau = \{ \varnothing, \mathbb{R} \} \cup \{ [a,\infty), (a, \infty) | a \in \mathbb{R} \}$$

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 $$\tau = \{ \varnothing, \mathbb{R} \} \cup \{ [a,\infty), (a, \infty) | a \in \mathbb{R} \}$$ 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 $$\tau = \{ \varnothing, \mathbb{R} \} \cup \{ [a,\infty), (a, \infty) | a \in \mathbb{R} \}$$ 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.