Real Analysis/Connected Sets

From Wikibooks, open books for an open world
< Real Analysis
Jump to: navigation, search


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.
Alternative Definition A set  X is called disconnected if there exists a continuous 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[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 \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[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 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).