Simply connected domain

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

Simply connected two-dimensional domains

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

Simply connected three-dimensional domains

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 Insighthttp://mathinsight.org/definition/simply_connected

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s