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 $$(0,0)$$.