Mr_Physics said:Not sure if singletons are open or not.
Mr_Physics said:An open ball of radius 1/2 I guess would just be a singleton, right?
An open subset in a metric space is a subset of the space that contains all of its interior points. This means that for any point in the subset, there exists a small ball around the point that is completely contained within the subset. In other words, every point in the subset has a neighborhood that is also contained within the subset.
A discrete metric is a type of distance function that is defined on a set of points and assigns a distance of 1 to any pair of distinct points, and a distance of 0 to any pair of identical points. In other words, the distance between any two points is either 0 or 1, depending on whether they are the same point or not.
To determine if a subset in a metric space is open, you can use the definition of an open subset: every point in the subset has a neighborhood that is also contained within the subset. This means that for any point in the subset, you can find a ball of a certain radius centered at that point that is completely contained within the subset. If this is true for all points in the subset, then the subset is open.
Yes, it is possible for a subset in a metric space to be both open and closed. This is known as a clopen set. In a discrete metric space, all subsets are both open and closed because every point has a neighborhood of radius 1 that is contained within the subset, making it open, and the complement of the subset is also a subset, making it closed.
In topology, the concept of open subsets is closely related to the concept of continuous functions. A function between two metric spaces is continuous if the preimage of any open subset in the codomain is an open subset in the domain. In other words, continuous functions preserve the openness of subsets. This means that if a function is continuous, then the inverse image of an open subset in the range will be an open subset in the domain.