# Homework Help: Topology generation

1. May 8, 2010

### complexnumber

1. The problem statement, all variables and given/known data

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

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

2. Relevant equations

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

3. 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 $$\EuScript{E}$$?

2. May 8, 2010

### jbunniii

Consider the sets

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

for $n = 1,2,\ldots$

What is the union of these sets?

3. May 8, 2010

### complexnumber

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.