Theorem 19.10.
If a topological space \(X\) is path connected, then \(X\) is connected.
Section Path Connectedness and Connectedness
Path connectedness and connectedness are different concepts, but they are related. In this section we will show that any path connected space must also be connected. We will see later that the converse is not true except in finite topological spaces.
Proof.
Suppose that \(X\) is path connected. Let \(a \in X\) and for any \(x \in X\) let \(p_x\) be a path from \(a\) to \(x\text{.}\) Let \(C_x = p_x([0,1])\text{.}\) Now \(C_x\) is the continuous image of the connected set \([0,1]\) in \(\R\text{,}\) so \(C_x\) is connected. Also, \(p_x(0) = a \in C_x\) and \(p_x(1) = x \in C_x\text{.}\) Thus, every set \(C_x\) contains \(a\) and so \(\bigcap_{x \in X} C_x\) is not empty. Therefore,
\begin{equation*}
C = \bigcup_{x \in X} C_x
\end{equation*}
is a connected subset of \(X\text{.}\) But every \(x \in X\) is in a \(C_x\text{,}\) so \(C = X\text{.}\) We conclude that \(X\) is connected.
In the following sections we explore the reverse implication in Theorem 19.10 — that is, does connectedness imply path connectedness.
