Connectedness

Connected Region

A region $D$ in $\mathbb{R}^n$ is connected iff any 2 points in $D$ can be joined by a piecewise smooth curve contained entirely in $D$.

Examples of connected regions:

• the open disk $x^2 + y^2 < 1$
• the whole plane
• the annulus $1 < x^2 + y^2 < 4$

Examples of regions that are not connected:

• 2 disjoint open disks
• the plane with the $y$-axis removed, since a point with $x > 0$ cannot be joined to a point with $x < 0$ without crossing the $y$-axis

Simply Connected Region

A connected open region $D$ in the plane is simply connected iff every simple closed curve contained in $D$ encloses only points of $D$.

In the plane, $D$ is simply connected iff it has no holes.

Examples of simply connected regions:

• the open disk $x^2 + y^2 < 1$
• the whole plane
• the upper half-plane $y > 0$

Examples of connected regions that are not simply connected:

• the annulus $1 < x^2 + y^2 < 4$, since a circle around the hole encloses points outside the annulus
• the punctured plane $\mathbb{R}^2 \setminus \{(0,0)\}$, since a circle around the origin encloses the origin