What is the precise definition of the open set? The definition I have been using until now has been that an open set is a set such that all of its points have some neighborhood that's contained in the set. The definition of neighborhood as far as I know is a collection of all the points within some given distance of a central point. Now I know this cannot be correct since we do not need a set to be a metric space in order to define a topology. But if a topology is a collection of open sets, then how can you have that without having a defined distance function? Wouldn't you need a distance function to define a neighborhood, and without a neighborhood how can you define an open set?