# Domains in .

Definition 2.1.1   Take and let be a positive real number. The open ball with center and radius is .

In Figure 1, we display the open unit disk, which is .

Definition 2.1.2   An open neighborhood of is a subset of containing an open ball centered at . The most used neighborhoods are the open balls themselves.

Example 2.1.3

• is a neighborhood of each of its points.
• The subset    Im is a neighborhhod of each of its points. If , then for    min , (see Figure 2).

Definition 2.1.4   A punctured neighborhood of contains all the points of a neighborhood of , excepted itself.

Example 2.1.5   is a punctured neighborhood of 0.

Definition 2.1.6   Let be a subset of . It is an open subset of if, for each point , there exists an open ball centered at and included in . .

In other words, is an open susbet of if, and only if, the following condition holds:

Example 2.1.7

• The set    Re is open. For any , we have    Re (see Figure 3(a)).
• The unit open disk is open: for any , take    min . Then 2.1 we have: (see Figure 3(b)).

Proposition 2.1.8   Let and be two open subsets of . Then the following hold:
1.
is open.
2.
is open.

Proof.
1.
Let . Suppose that ; as is open, there exists R>0 such that , whence . If , a similar argument works, whence the result.
2.
Let . As and are open, there exist two positive numbers and such that and . Take . Thus and we are done.

Remark 2.1.9

1.
Proposition 1.8 can be generalized to the union and the intersection of any finite number of open subsets of . The proof (by induction) is left to the reader.
2.
For an infinite family of open subsets of , the intersection cannot be open. For example. take , for any complex number . Then , and it is easy to show that a set which contains only a single point is not open.

Definition 2.1.10   A closed set is the complement of an open set.

Example 2.1.11   The closed unit-disk is a closed set, as its complementary set is open: it is (see Figure 5). For any , i.e. for any such that , let ; then we have .

The following proposition will be very useful throughout our study of analytic functions (chapter chapter analytic functions) and further.

Proposition 2.1.12   Let be a finite set of points in . Then is open.

Proof. Denote and take any . Now denote (i.e. is the distance between the images in the palne of the complex numbers and ). The set is a finite set of positive real numbers, thus it has a minimal element, say . Using the triangle inequality, it is easy to show that .

Proposition 2.1.13   Let and be two closed subsets of . Then the following hold:
1.
is closed.
2.
is closed.

The proof is left to the reader, using De Morgan laws.

Remark 2.1.14   There exist subsets of the complex plane which are neither open nor closed. For example, take the set    Re (Figure 6).

If , then but every open ball centered at contains points which do not belong to . Thus is not open.

Now consider the complementary subset of in the and take . Then , but every open ball centered at contains points which do not belong to . Thus is not open, whence is not closed.

Definition 2.1.15   A connected set is a subset of such that any two points in the set can be connected by a path of straight segments totally contained in the set. For example,
1. The punctured plane is connected and without a whole straight line is not connected.
2. The subset is not connected. See Figure 7: it is impossible to connect the points and with a finite sequence of segments totally included in .

A domain is an open connected set.

Example 2.1.16

• The unit disk is a domain (see Figure 8(a)).
• The whole of is a domain.
• The subset    Re is not open, thus it is not a domain (see Figure 8(b)).

Definition 2.1.17   A boundary point of a set is a point in such that every ball centered at contains at least one point of and at least one point not in . The set of all the boundary points of is called the boundary of .

Example 2.1.18   Let . Then every point on the line whose equation is is a boundary point.

Let , where . Then, for any , the point is in the open ball , but not in . Thus is a boundary point of .

Definition 2.1.19   An interior point of a set is a point such that there exists an open ball centered at and totally contained in . An exterior point of a set is a point such that there exists an open ball centered at and all of whose points are out of (see Fig. 9(b)).

Example 2.1.20   Let (see Figure 10). A point such that or such that is a point exterior to :
• If , take    min ; then .
• If , take ; then .

A point such that is an interior point; take    min , then .

Definition 2.1.21   A bounded set is a set for which there exists a positive number such that (i.e. it is composed only of interior points of a certain circle centered at 0).

Example 2.1.22

• The annulus is bounded. For example, it is a subset of the disk, whose center is at the origin and whose radius is equal to (see Figure 11(a)).
• The whole of is not bounded.
• The set    Re is not bounded. For every , there are complex numbers whose real part has a value between and and whose absolute value is greater than , e.g. (see Figure 11(b)).

Remark 2.1.23   There is no connection between the notions of a closed set (definition 1.10) and of a bounded set (defintion 1.21). We mean a set can have either both properties, or only one of them, or none of them. For example:
(i)
The set is closed and bounded.
(ii)
The unit ball is bounded nut not closed.
(iii)
The set is closed but unbounded.
(iv)
The set is neither closed nor bounded.

Noah Dana-Picard 2007-12-24