Why is empty set an open set? I thought about it for a long time, but I can't come up some good ideas. I think that empty set has no elements,how to use the definition of an open set to prove the proposition.
The definition of an open set: a set S in n-dimensional space is called open if all its points are interior points.
It's vacuously true. All points $\in\emptyset$ are interior points, have a blue-eyed pet unicorn, and live in Surrey.
Consider the following definition of an open set: For every element in A, there exists a ball of the element contained in A. What is the negation of that statement? Well, it is: There exists an element in A such that every ball is not contained in A. Clearly, the empty set does not satisfy the negation of an open set, therefore it must be vacuously true that the empty set satisfies the definition of an open set.