$\mathbb{R},\emptyset$, and every interval $(-n,n)$ are a topology

Question

I am taking a course in general topology. And I am struggling with the following exercise:

Let $\mathbb{R}$ be the set of all real numbers. Prove that each of the following subsets of $\mathbb{R}$ is a topology.

i) $\mathscr{T}$ consists of $\mathbb{R},\emptyset$, and every interval $(-n,n)$, for $n$ any positive integer.

Definition of topology:

1. $X,\emptyset\in\mathscr{T}$
2. If $\{x_\lambda:\lambda\in\Lambda\}\subseteq\mathscr{T}$, so that $\bigcup_{\lambda\in\Lambda}^{}x_\lambda\in\mathscr{T} $
3. If $A,A'\in\mathscr{T}$ then $A\cap A\in\mathscr{T}$

1) The first definition is checked once $\mathbb{R},\emptyset\in\mathscr{T}$

2) The second definition is the one I do not know how to tackle. It seems straight-forward to mw that the union of any $(-n,n)$ belongs in the topology. However I do not know how to write a proof about it.

Question:

How should I write a proof of the second axiom?

Answer 1

The core of your proof ought to be that

• if $\Lambda = \{\lambda_1, \lambda_2, … \lambda_n \}$ is a non-empty finite set of (positive) integers, then

$$\bigcup_{\lambda \in \Lambda}x_{\lambda} = x_{\max \Lambda}$$

and that

• if $\Lambda$ is infinite then the union is $\mathbb{R}$.

Answer 2

Actually $$\bigcup (-n,n)=(-\max n,\max n)$$ if that the maximum exists, otherwise $$\bigcup (-n,n)=\mathbb R$$