How to state following sentence formally?
Let $\tau$ be family of sets (possibly infinite). Any finite intersection of sets in $\tau$ is in $\tau$.
My attempt is: $\tau$ is family of sets and $O_1\in\tau \wedge O_2\in\tau \Rightarrow O_1\cap O_2\in\tau$.
Basically I don't know how do define:
- Intersection of finite number of sets in $\tau$.
- Intersection of possibly infinite number of sets in $\tau$.
I need the definition of finite intersection to formally define an Open Set in Topology theory.
Your statement is fine. It says that whenever you take two sets in your family and intersect them, the result is still in your family. This is equivalent to saying that any finite intersection of sets in your family is still in your family, since we can do the finite number of intersections one at a time: $$O_1 \cap O_2 \cap \ldots \cap O_n = (((O_1 \cap O_2)\cap O_3) \ldots ) \cap O_n$$ Notice that each of the intersections on the right is actually just the intersection of two sets.
To talk about an intersection of an infinite number of sets, we typically do the following. Let $\{O_i\}_{i \in I} \subset \tau$, where $I$ is some indexing set (it can have any cardinality, hence it accounds for any size of intersection). Then we denote the intersection of these sets as $$\bigcap_{i \in I} O_i = \{e : e \in O_i \text{ for every } i \in I\}$$ Thus we can make reference to an infinite intersection of sets by referring to $\bigcap_{i \in I} O_i$.