# 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 ${\displaystyle \mathbb {R} }$ should be connected, but a set ${\displaystyle A}$ consisting of two disjoint closed intervals ${\displaystyle [a,b]}$ and ${\displaystyle [c,d]}$ should not be connected.

Definition A set in ${\displaystyle A}$ in ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle X}$ is called disconnected if there exists a continuous function ${\displaystyle f:X\to \{0,1\}}$, such a function is called a disconnection. If no such function exists then we say ${\displaystyle X}$ is connected.
Examples The set ${\displaystyle [0,2]}$ cannot be covered by two open, disjoint intervals; for example, the open sets ${\displaystyle (-1,1)}$ and ${\displaystyle (1,2)}$ do not cover ${\displaystyle [0,2]}$ because the point ${\displaystyle x=1}$ is not in their union. Thus ${\displaystyle [0,2]}$ is connected.
However, the set ${\displaystyle \{0,2\}}$ can be covered by the union of ${\displaystyle (-1,1)}$ and ${\displaystyle (1,3)}$, so ${\displaystyle \{0,2\}}$ 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 ${\displaystyle \mathbb {R} ^{2}\setminus \{(0,0)\}}$. 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, ${\displaystyle \mathbb {R} \setminus \{0\}}$ is not path-connected, because for ${\displaystyle a=-3}$ and ${\displaystyle b=3}$, there is no path to connect a and b without going through ${\displaystyle x=0}$.

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 ${\displaystyle \mathbb {R} ^{n}}$ with ${\displaystyle n>1}$. 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 ${\displaystyle A}$ is simply-connected if any loop completely contained in ${\displaystyle A}$ can be shrunk down to a point without leaving ${\displaystyle A}$.

An example of a Simply-Connected set is any open ball in ${\displaystyle \mathbb {R} ^{n}}$. However, the previous path-connected set ${\displaystyle \mathbb {R} ^{2}\setminus \{(0,0)\}}$ 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 ${\displaystyle (0,0)}$.