No. The complement of an open set is an closed set but the complement of a "ball" is not a "ball". An open ball is of the form [itex]B_r(p)= \{ q| d(p, q)< r\right}[/itex]. In R, an "open ball" is an open interval, (a, b). Its complement is [itex](-\infty, a]\cup [b, \infty)[/itex] which is closed but not a "ball".sampahmel said:Homework Statement
I am using Rosenlicht's Intro to Analysis to self-study.
1.) I learn that the complements of an open ball is a closed ball.
I don't see what one has to do with the other. The complement of any open set is closed, the complement of any closed set is open. The complement of a set that is neither closed nor open is neither closed nor open. The "half open interval" in R, (0, 1], is neither closed nor open.And...
2.) Some subsets of metric space are neither open nor closed.
Homework Equations
Is something amiss here? I do not understand how both can be true at the same time.
An open set in a metric space is a subset of the metric space where every point in the set has an open neighborhood contained within the set. This means that for every point in the set, there is a radius around that point within which all other points are also in the set.
A set is open in a metric space if all of its points have an open neighborhood contained within the set. A set is closed in a metric space if it contains all of its limit points, meaning all the points that can be approached arbitrarily closely from within the set.
Yes, a set can be both open and closed in a metric space if it is the entire space or the empty set. In other words, the only sets that are both open and closed in a metric space are the whole space and the empty set.
In general, a function is continuous if and only if the preimage of any open set is an open set. This means that open sets are important for defining continuity, while closed sets can be used to define discontinuity.
In a metric space, a set is closed if and only if its complement is open. This means that open and closed sets are complementary to each other in a metric space, and both are necessary for defining the topology of the space.