(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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]

2. Relevant 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]

3. 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]?

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# Homework Help: Topology generation

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