Is there an example of family of open intervals in $\mathbb R$ …?

Question

Is there an example of family of open intervals in $\mathbb R$ such that any arbitrary union of such open intervals is again an open interval? In other words can we define a topology on $\mathbb R$ with open intervals only?, (i.e. $A$ is open in $\mathbb R$ if and only if $A$ is an open interval) However we have such a topology when we take such collection as base.

Answer 1

Your two questions are not really equivalent; under the usual topology, the collection of open intervals $\{(-r,r)\mid r\in\mathbb{R}\}$ and $\mathbb{R}$ satisfies your first condition (interpret $(-r,r)$ with $r\lt 0$ as empty): an arbitrary union of such open intervals is again an open interval. It even defines a topology, but it is not a topology such all open intervals are open.

No topology can consist exactly of the open intervals, since it would necessarily contain $(0,1)$ and $(2,3)$ as open sets, hence contain its union as an open set, but $(0,1)\cup(2,3)$ is not an interval.

Answer 2

For your second question, the answer is NO. If it satisfies your condition, it can't generate a topology. Just as Arturo Magidin said, $(0,1)$ and $(2,3)$ as open sets, hence contain its union as an open set, but $(0,1)\cup(2,3)$ is not an interval:)

Answer 3

(I'll approach this somewhat differently.)

A family of open intervals in $\mathbb{R}$ such that the union of any subfamily is again an open interval certainly exists: take, for example, the family of all intervals of the form $(r,s)$ where $r < 0 < s$.

Suppose now that $\mathcal{I}$ is such a family of open intervals which moreover forms a topology on $\mathbb{R}$. I claim that this topology is not T$_1$:

Assume that $\{ x \}$ is closed in this topology for some $x \in \mathbb{R}$. Then there must be an $I \in \mathcal{I}$ such that $x-1 \in I \subseteq ( - \infty , x )$, and similarly there must be an $J \in \mathcal{I}$ such that $x+1 \in J \subseteq (x , + \infty )$. However it easily follows that $I \cup J$ is not an interval!