Which of the following are elements of ?
Which of the following are interior points of ?
Which of the following are boundary points of ?
We generalize the notion of open and closed intervals to open and closed sets in .
Given a set , we say that is an element of if is one of the distinct objects in , and we write to denote this.
Given two sets and , we say that is a subset of if every element of is also an element of write to denote this.
With these ideas in mind, we now discuss special types of subsets.
We give these definitions in general, for when one is working in since they are really not all that different to define in than in .
An open ball in centered at with radius is the set of all points such that the distance between and is less than .
In an open ball is often called an open disk.
Intuitively, is an interior point of if we can squeeze an entire open ball centered at within .
Which of the following are elements of ?
Which of the following are interior points of ?
Which of the following are boundary points of ?
We can now generalize the notion of open and closed intervals from to open and closed sets in .
Intuitively, this means that we can enclose all of the set within a large enough ball centered at the origin.
Let’s now look at a few examples.
This is the region bounded by the ellipse . Since the region includes the boundary (indicated by the use of “”), the set containsdoes not contain all of its boundary points and hence is closed. The region is boundedunbounded as a disk of radius , centered at the origin, contains .