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 .

A **set** is a collection of distinct objects.

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.

(Open Balls)

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**.

(Interior and Boundary Points) Suppose that .

- A point is an
**interior point**of if there exists an open ball .Intuitively, is an interior point of if we can squeeze an entire open ball centered at within .

- A point is a
**boundary point**of if all open balls centered at contain both points in and points not in . - The
**boundary**of is the set that consists of all of the boundary points of .

Suppose that .

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 .

(Open and Closed Sets)

- A set is
**open**if every point in is an interior point. - A set is
**closed**if it contains all of its boundary points.

(Bounded and Unbounded)

- A set is
**bounded**if there is an open ball such thatIntuitively, this means that we can enclose all of the set within a large enough ball centered at the origin.

- A set that is not bounded is called
**unbounded**.

Let’s now look at a few examples.

Consider the function .

- The domain of the function is the set of all for which , which we can
write in set
- The point is an interior pointa boundary pointnot an element of .
- The point is an interior pointa boundary pointnot an element of .
- The domain is openclosedneither open nor closed and boundednot bounded .

We’ve already found the domain of this function to be

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 .

Determine if the domain of is open, closed, or neither, and if it is bounded.

As we
cannot divide by , we find the domain to be In other words, the domain is the set of
all points *not* on the line . For your viewing pleasure, we have included a graph:
Note how we can draw an open disk around any point in the domain that lies entirely
inside the domain, and also note how the only boundary points of the domain are the
points on the line . We conclude the domain is an open seta closed setneither
open nor closed set
. Moreover, the set is boundedunbounded
.