## What is connected and simply connected?

A pathwise-connected domain is said to be simply connected (also called 1-connected) if any simple closed curve can be shrunk to a point continuously in the set. If the domain is connected but not simply, it is said to be multiply connected.

### What is a simple connected region?

A region is simply connected if every closed curve within it can be shrunk continuously to a point that is within the region. In everyday language, a simply connected region is one that has no holes.

#### How do you determine if a set is simply connected?

A region D is said to be simply connected if any simple closed curve which lies entirely in D can be pulled to a single point in D (a curve is called simple if it has no self intersections).

**Is the annulus 1 <| z |< 2 Simply Connected?**

Yes, this is simply connected. Every loop can be continuously deformed to a point without leaving the strip. There are no holes in the strip. lies in the annulus but cannot be shrunk to a point without leaving the annulus.

**Can a region be simply connected but not connected?**

Informally, an object in our space is simply connected if it consists of one piece and does not have any “holes” that pass all the way through it. Spaces that are connected but not simply connected are called non-simply connected or multiply connected.

## Is a domain simply connected?

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.

### Is plane simply connected?

Definition A domain D is called simply connected is every closed contour Γ in D can be continuously deformed to a point in D. The whole complex plane C and any open disk Br (z0) are simply connected.

#### What makes a domain simply connected?

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. For two-dimensional regions, a simply connected domain is one without holes in it. A simply connected domain is one without holes going all the way through it.

**Is r3 without origin simply connected?**

So our region is all of R^3 except the origin. And in two-dimensional space, this was not simply connected. But in three-dimensional space it is simply connected. So actually, this region, even though in two-dimensional space it was not simply connected, in three-dimensional space it is.

**Is annulus simply connected domain?**

Definition A domain D is called simply connected is every closed contour Γ in D can be continuously deformed to a point in D. The whole complex plane C and any open disk Br (z0) are simply connected. We’ll see shortly that the annulus A = {z ∈ C : 1 < |z| < 2} is not simply connected.

## Is annulus a domain?

A doubly-connected planar domain between two closed Jordan curves without common points, one of each encloses the other.

### IS SO 2 simply connected?

SO(2) is path-connected but not simply connected, that is, there is a closed path in SO(2) that cannot be continuously shrunk to a point. R is path-connected and simply connected. Another difference is that both O(2) and SO(2) are compact, that is, closed and bounded, and R is not.