That's correct. Roughly speaking a set is closed if the set contains the boundary of that set.Fairy111 said:
In words it means the set of all z such that the magnitude of z is less than one. Geometrically, what does this set look like in the complex plane?Fairy111 said:
Yes I did, thanks.Fairy111 said:
You're on the right lines, the set contains all points in the complex plane who's magnitude is less than two. So yes, this does look like a circle of radius two, but the question is: does the set include the boundary (i.e. the points on the circumference) of the circle?Fairy111 said:
Sounds good to meFairy111 said:
An open set is a set of points in a metric space that does not include its boundary points. In other words, for every point in an open set, there exists a neighborhood around that point that is also contained within the set.
A closed set includes its boundary points, while an open set does not. Additionally, a closed set is the complement of an open set, meaning that it contains all points not included in the open set.
To determine if a set is open or closed, you can look at its boundary points. If the set includes its boundary points, it is closed. If the set does not include its boundary points, it is open.
This notation represents the absolute value of the complex number z being less than 2. This can be represented geometrically as a circle with a radius of 2 centered at the origin on the complex plane.
Understanding open and closed sets is important in various areas of mathematics and science, including topology, analysis, and probability theory. It allows for a more precise and rigorous definition of concepts such as continuity, convergence, and connectedness. Additionally, it can be applied in various real-world scenarios, such as in modeling physical systems or analyzing data.