4.3 Path-Connectedness
A concept closely related to connectedness is that of path-connectedness, which we define as follows:
Definition 4.3.1. Let \(X\) be a topological space. A path in \(X\) from \(x_0\) to \(x_1\) is defined to be a continuous
function \(\gamma :[0,1]\rightarrow X\) such that \(\gamma (0)=x_0\) and \(\gamma (1)=x_1\).
A topological space is said to be path-connected if and only if, given any two points \(x_0\) and \(x_1\) in \(X\),
then there exists a path in \(X\) from \(x_0\) to \(x_1\).
Proof. Let X be a path-connected topological space and let \(f:X\rightarrow Z\) be continuous integer-valued function on X. If \(x_0,x_1\) are any two points of X, there exists a path \(\gamma :[0,1]\rightarrow X\) such that \(\gamma (0)=x_0\) and \(\gamma (1)=x_1\). But then \(fo\gamma :[0,1]\rightarrow Z\) is continuous integer valued function on [0,1]. But [0,1] is connected, and so \(fo\gamma \) is constant. It follows that \(f(x_0)=f(x_1)\). Thus every integer valued function on X is constant. Therefore X is connected. □
Example 4.3.3. The topological space \((\mathbb {R},\tau )\), where \(\tau \) is the topology arising from the usual metric is path-connected and therefore connected.
Proof. Let \(x_0\) and \(x_1\) be any two fixed points in \(\mathbb {R}\) and define \(\gamma :[0,1]\rightarrow \mathbb {R}\) by \(\gamma (t)=(1-t)x_0+tx_1\) and since \(\gamma \) is a linear function of t it is continuous. Thus \(\gamma \) is a path in \(\mathbb {R}\) connecting \(x_0\) and \(x_1\) and so \(\mathbb {R}\) is connected. □
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.