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, both of which it intersects.
- 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.
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.
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 .