Regions in the Complex plane:
Neighborhood: A neighborhood consists of all points z lying inside but not on a circle centered at zo and with a specified positive radius ε. When the value of ε is understood or is immaterial in the discussion, the set (1) is often referred to as just a neighborhood.
\[\left | z-z_{0} \right |< \epsilon .........(1)\]
Deleted Neighborhood: A deleted neighborhood or punctured disk consisting of all points z in an ε neighborhood of z0 except for the point z0 itself.
\[0 < \left | z-z_{0} \right |< \epsilon ......(2)\]
Interior Point: A point z0 is said to be an interior point of a set S whenever there is some
neighborhood of z0 that contains only points of S.
Exterior Point: A point z0 is called an exterior point of S when there exists a neighborhood of it containing no points of S.
Boundary Point: A point z0 is called a boundary point if it is neither an interior or exterior point of a set S. A boundary point is, therefore, a point all of whose neighborhoods contain at least one point in S and at least one point not in S.
Boundary: The totality of all boundary points is called the boundary of S. The circle |z| = 1,
for instance, is the boundary of each of the sets
(3) |z| < 1 and |z| ≤ 1.
Open Set: A set is open if it contains none of its boundary points. Ex: |z| < 1 is an open set.
Closed Set: A set is closed if it contains all of its boundary points.
Closure of a Set: The closure of a set S is the closed set consisting of all points in S together with the boundary of S. Ex: |z| ≤ 1 is closure of |z| < 1
Neither Open nor Closed set: For a set to be not open, there must be a boundary point that is contained in the set; and if a set is not closed, there exists a boundary point not contained in the set. Observe that the punctured disk 0 < |z| ≤ 1 is neither open nor closed.
Both Open and Closed set: A set which has no boundary points. The set of all complex numbers is, on the other hand, both open and closed since it has no boundary points.
Connected Set: An open set S is connected if each pair of points z1 and z2 in it can be joined by a polygonal line, consisting of a finite number of line segments joined end to end, that lies entirely in S. The open set |z| < 1 is connected. The annulus 1 < |z| < 2 is, of course, open and it is also connected.
Domain: A nonempty open set that is connected is called a domain. Note that any neighborhood is a domain.
Region: A domain together with some, none, or all of its boundary points is referred to as a
region.
Bounded and Unbounded Set: A set S is bounded if every point of S lies inside some circle |z| = R; otherwise, it is unbounded. Both of the sets (3) are bounded regions, and the half plane Re z ≥ 0 is unbounded.
Accumulation Point: A point z0 is said to be an accumulation point of a set S if each deleted neighborhood of z0 contains at least one point of S.
A Closed Set Contains All of its Accumulation Points: It follows that if a set S is closed, then it contains each of its accumulation points. For if an accumulation point z0 were not in S, it would be a boundary point of S; but this contradicts the fact that a closed set contains all of its boundary points. Thus a set is closed if and only if it contains all of its accumulation points.
Evidently, a point z0 is not an accumulation point of a set S whenever there exists some deleted neighborhood of z0 that does not contain at least one point of S.
*Reference : Brown & Churchill - Complex Variables and Application
Deleted Neighborhood: A deleted neighborhood or punctured disk consisting of all points z in an ε neighborhood of z0 except for the point z0 itself.
\[0 < \left | z-z_{0} \right |< \epsilon ......(2)\]
neighborhood of z0 that contains only points of S.
Exterior Point: A point z0 is called an exterior point of S when there exists a neighborhood of it containing no points of S.
Boundary Point: A point z0 is called a boundary point if it is neither an interior or exterior point of a set S. A boundary point is, therefore, a point all of whose neighborhoods contain at least one point in S and at least one point not in S.
Boundary: The totality of all boundary points is called the boundary of S. The circle |z| = 1,
for instance, is the boundary of each of the sets
(3) |z| < 1 and |z| ≤ 1.
Open Set: A set is open if it contains none of its boundary points. Ex: |z| < 1 is an open set.
Closed Set: A set is closed if it contains all of its boundary points.
Closure of a Set: The closure of a set S is the closed set consisting of all points in S together with the boundary of S. Ex: |z| ≤ 1 is closure of |z| < 1
Neither Open nor Closed set: For a set to be not open, there must be a boundary point that is contained in the set; and if a set is not closed, there exists a boundary point not contained in the set. Observe that the punctured disk 0 < |z| ≤ 1 is neither open nor closed.
Both Open and Closed set: A set which has no boundary points. The set of all complex numbers is, on the other hand, both open and closed since it has no boundary points.
Connected Set: An open set S is connected if each pair of points z1 and z2 in it can be joined by a polygonal line, consisting of a finite number of line segments joined end to end, that lies entirely in S. The open set |z| < 1 is connected. The annulus 1 < |z| < 2 is, of course, open and it is also connected.
Domain: A nonempty open set that is connected is called a domain. Note that any neighborhood is a domain.
Region: A domain together with some, none, or all of its boundary points is referred to as a
region.
Bounded and Unbounded Set: A set S is bounded if every point of S lies inside some circle |z| = R; otherwise, it is unbounded. Both of the sets (3) are bounded regions, and the half plane Re z ≥ 0 is unbounded.
Accumulation Point: A point z0 is said to be an accumulation point of a set S if each deleted neighborhood of z0 contains at least one point of S.
A Closed Set Contains All of its Accumulation Points: It follows that if a set S is closed, then it contains each of its accumulation points. For if an accumulation point z0 were not in S, it would be a boundary point of S; but this contradicts the fact that a closed set contains all of its boundary points. Thus a set is closed if and only if it contains all of its accumulation points.
Evidently, a point z0 is not an accumulation point of a set S whenever there exists some deleted neighborhood of z0 that does not contain at least one point of S.
*Reference : Brown & Churchill - Complex Variables and Application
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