Simply connectedness

Informally, a thick 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.

A sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on the surface of a sphere can contract to a point, even though it has a "hole" in the hollow center.

The stronger condition, that the object has no holes of any dimension, is called contractibility.

Another characterization of simply-connectedness is the following:

\(X\) is simply-connected if and only if 

• it is path-connected, and 
• whenever \(p: [0,1] \rightarrow X\) and \(q: [0,1] \rightarrow X\) are two paths (i.e. continuous maps) with the same start and endpoint (\(p(0)=q(0)\) and \(p(1) == q(1)\)), then \(p\) and \(q\) are homotopic relative to {0,1}.

Intuitively, this means that \(p\) can be "continuously deformed" to get \(q\) while keeping the endpoints fixed. Hence the term simply connected: for any two given points in \(X\), there is one and "essentially" only one path connecting them.

Examples:

• All convex sets in \(\mathbb{R}^n\) are simply connected.
• A sphere is simply connected.