Applejacks said:Homework Statement
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Homework Equations
A simply connected domain D in the complex plane is an open and path
connected set such that every simple closed path in D encloses only points of D.
The Attempt at a Solution
The answers are a,c and d.I don't understand why they all aren't simply connected. They are all path connected and open. Am I misunderstanding the definition of open?
Dick said:There's a path in the annulus in b that encloses the origin z=0. What is it? Is z=0 in the annulus?
Deveno said:there's a lot of such paths. even discounting homotopic ones, there's still an infinite number.
Deveno said:and a good one, at that! :)
A simply connected domain is a type of mathematical region in which any closed curve can be continuously shrunk to a single point without leaving the region. In other words, there are no holes or gaps in the domain.
A multiply connected domain has one or more holes or gaps, while a simply connected domain does not. This means that in a simply connected domain, any closed curve can be shrunk to a single point, while in a multiply connected domain, there are closed curves that cannot be shrunk to a point without leaving the domain.
Some examples of simply connected domains include a disk, a rectangle, and a sphere. These are all regions that do not have any holes or gaps and any closed curve can be continuously shrunk to a point within the region.
The concept of a simply connected domain is important in complex analysis and topology. In complex analysis, simply connected domains are used to define and study holomorphic functions, which are important in many areas of mathematics and physics. In topology, simply connected domains are used to study the properties of continuous functions and surfaces.
Yes, a simply connected domain can be unbounded, meaning that it extends infinitely in one or more directions. For example, a plane or a cylinder are both simply connected domains that are unbounded. The key characteristic of a simply connected domain is that it does not have any holes or gaps, regardless of its size or shape.