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

[complexnumber]

Homework Statement

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

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

Homework Equations

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

The Attempt at a Solution

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

 

[jbunniii]

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

Consider the sets

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

for [itex]n = 1,2,\ldots[/itex]

What is the union of these sets?

 

[complexnumber]

Consider the sets

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

for [itex]n = 1,2,\ldots[/itex]

What is the union of these sets?

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

Thanks very much for your help.