Open Balls and Open Sets An open ball is a set of points in a metric space that is completely contained within the space. In simpler terms, it is a set of p...
Open Balls and Open Sets
An open ball is a set of points in a metric space that is completely contained within the space. In simpler terms, it is a set of points that can be reached from the origin (the point at the center of the space) by traversing only a finite number of steps along the lines of the space.
For example, the open ball centered at the point (1, 2) in the plane would be represented by the set of all points (x, y) such that |x - 1| + |y - 2| < 1.
Open Sets
An open set is a set that is completely contained within another set. In other words, every point in the open set can be reached from the boundary of the set by traversing only a finite number of steps.
For instance, the open set containing the point (3, 4) would be represented by the set of all points (x, y) such that x > 2 and y > 3.
Convergence
When a sequence of points converges to a single point, it does so in a sequence of steps. The sequence has a unique limit point. In other words, the points in the sequence get closer and closer to the limit point as they are taken further apart.
For example, consider the sequence of points (1, 2), (2, 4), (3, 6), and so on. The sequence converges to the point (1, 2) since the distance between consecutive points approaches zero as the sequence progresses