Real Analysis/Connected Sets

Intuitively, the concept of connectedness is a way to describe whether sets are "all in one piece" or composed of "separate pieces". For motivation of the definition, any interval in $\mathbb {R}$ should be connected, but a set $A$ consisting of two disjoint closed intervals $[a,b]$ and $[c,d]$ should not be connected.

Definition A set in $A$ in $\mathbb {R} ^{n}$ is connected if it is not a subset of the disjoint union of two open sets, both of which it intersects.

Alternative Definition A set $X$ is called disconnected if there exists a continuous, surjective function $f:X\to \{0,1\}$, such a function is called a disconnection. If no such function exists then we say $X$ is connected.

Examples The set $[0,2]$ cannot be covered by two open, disjoint intervals; for example, the open sets $(-1,1)$ and $(1,2)$ do not cover $[0,2]$ because the point $x=1$ is not in their union. Thus $[0,2]$ is connected.

However, the set $\{0,2\}$ can be covered by the union of $(-1,1)$ and $(1,3)$, so $\{0,2\}$ is not connected.

Path-Connected

A similar concept is path-connectedness.

Definition A set is path-connected if any two points can be connected with a path without exiting the set.

A useful example is $\mathbb {R} ^{2}\setminus \{(0,0)\}$. Any two points a and b can be connected by simply drawing a path that goes around the origin instead of right through it; thus this set is path-connected. However, $\mathbb {R} \setminus \{0\}$ is not path-connected, because for $a=-3$ and $b=3$, there is no path to connect a and b without going through $x=0$.

As should be obvious at this point, in the real line regular connectedness and path-connectedness are equivalent; however, this does not hold true for $\mathbb {R} ^{n}$ with $n>1$. When this does not hold, path-connectivity implies connectivity; that is, every path-connected set is connected.

Simply Connected

Another important topic related to connectedness is that of a simply connected set. This is an even stronger condition that path-connected.

Definition A set $A$ is simply-connected if any loop completely contained in $A$ can be shrunk down to a point without leaving $A$.

An example of a Simply-Connected set is any open ball in $\mathbb {R} ^{n}$. However, the previous path-connected set $\mathbb {R} ^{2}\setminus \{(0,0)\}$ is not simply connected, because for any loop p around the origin, if we shrink p down to a single point we have to leave the set at $(0,0)$.