LINE INTEGRALS AND GREEN’S THEOREM 
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exclusive of the positive axis of x, — all these are examples of simply 
connected regions. 
A circular ring is an example of a multiply connected region. 
Consider a region S lying inside a curve C, but outside each of the 
curves Ci, • ••, C n . If cuts he made along lines joining the inner 
boundaries with the outer boundary, the new 
region, S', will be simply connected. It is clear 
that n such cuts suffice. These may be drawn 
in a variety of ways. Thus the curves C u • • •, C n 
could be connected in series, and one of them „ 
connected also with C. But it can be shown 
that, no matter how the cuts be drawn, their number will always be 
the same, namely, n. Such a region is called doubly (n = 2) or triply 
(n = 3) or n-tuply connected. 
A simply connected region cannot have a boundary that consists 
of more than a single piece. But not all regions whose boundary 
consists of a single piece are simply connected. Thus the exterior 
of a circle is multiply connected. It is said to be doubly connected, 
since a single cut, as the ray which consists in a radius produced, 
would yield a simply connected region. Again, the whole plane 
with the exception of a single point is a doubly connected region. 
A simply connected region can also be characterized by the fact 
that any closed curve drawn in the region can be deformed continu 
ously (like a flexible elastic string) to an interior point of the region 
— more properly, until it lies wholly within an arbitrarily small 
neighborhood of the point — without ever coming into collision with 
the boundary of the region. 
Space of Three Dimensions. The ideas and definitions just set 
forth admit a two-fold generalization to space of three dimensions. 
Consider the space V between two concentric spheres. In this shell 
a surface can be drawn (namely, a third concentric sphere) which 
contains a part of the boundary of V in its interior. Thus we should 
be led to consider V as multiply connected. But a closed curve 
drawn in V can be deformed continuously to an interior point of V 
— i.e. until it lies wholly within an arbitrarily small neighborhood 
of the point — without ever touching the boundary of V. For this 
reason it is natural to regard V as simply connected. We can meet 
both situations by saying that V is linearly simply connected, but is 
multiply connected with respect to surfaces. 
The space (either interior or exterior) bounded by an anchor ring