A simply connected domain is a path-connected domain where one can continuously shrink any simple closed curve into a point while remaining in the domain.

The unit disc {(z ∈ C||z| 6 1} including the boundary.

This is not an open set, so is not a region and hence cannot be a simply connected region.

The unit disc {(z ∈ C||z| < 1}.The set is a region. Using the first definition, clearly it is simply connected because if we place any loop in D, it can be pulled to a point. Using the second definition, we can connect any point z in Dc to ∞ by taking a radial line from z outward i.e. γ(t) = z + zt

For two-dimensional regions, a simply connected domain is one without holes in it.

For three-dimensional domains, the concept of simply connected is more subtle. A simply connected domain is one without holes going all the way through it. However, a domain with just a hole in the middle (like a ball whose center is hollow) is still simply connected, as we can continuously shrink any closed curve to a point by going around the hole and remaining in the domain. On the other hand, a ball with a hole drilled all the way through it, or a spool with a hollow central axis, is not simply connected. A closed curve that went around the hole could not be shrunk to a point while remaining in the domain. There is no way for the curve to bypass the the hole so it remains stuck around it.

http://faculty.up.edu/wootton/Complex/Chapter8.pdf

Nykamp DQ, “Simply connected definition.” From *Math Insight*. http://mathinsight.org/definition/simply_connected

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