Can you please post your progress on this question or anything you have tried? Start with defining open sets in an arbitrary metric space.eraldcoil said:Let \(\displaystyle (E=]-1,0]\cup\left\{1\right\},d) \) metric space with \(\displaystyle d\) metric given by \(\displaystyle d(x,y)=|x-y|\), and \(\displaystyle ||\)absolute value.
How I can find open sets of E explicitly?
Thanks in advance.
An open set in a metric space is a set of points where every point has a neighborhood contained entirely within the set. This means that for every point in the set, there is a small radius around it that only includes other points within the set.
Open sets and closed sets are complementary concepts. A closed set contains all of its limit points, while an open set does not contain any of its limit points. In other words, a point on the boundary of a closed set is also considered part of the set, while a point on the boundary of an open set is not.
Open sets are fundamental to the study of topology, as they allow for the definition of key concepts such as continuity and convergence. They also provide a basis for defining more complex sets and functions in a metric space.
Open sets have a wide range of applications in various fields, including physics, engineering, and computer science. For example, in physics, open sets are used to define the continuity of space-time in general relativity. In computer science, open sets are used in algorithms for image processing and pattern recognition.
Yes, an open set can contain an infinite number of points. In fact, many open sets in metric spaces are uncountable, meaning they contain an infinite number of points that cannot be listed or counted in a finite amount of time. This is because the concept of open sets is not limited to finite sets, but can also include infinite sets.