# 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 should be connected, but a set consisting of two disjoint closed intervals and should not be connected.

**Definition**A set in in is**connected**if it is not a subset of the disjoint union of two open sets.**Alternative Definition**A set is called**disconnected**if there exists a continuous function , such a function is called a**disconnection**. If no such function exists then we say is**connected**.

**Examples**The set cannot be covered by two open, disjoint intervals; for example, the open sets and do not cover because the point is not in their union. Thus is connected.- However, the set can be covered by the union of and , so is
*not*connected.

## Path-Connected[edit]

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 . 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, is *not* path-connected, because for and , there is no path to connect a and b without going through .

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 with . When this does not hold, path-connectivity implies connectivity; that is, every path-connected set is connected.

## Simply Connected[edit]

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

An example of a Simply-Connected set is any open ball in . However, the previous path-connected set 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 .