Definition 19.1.
Let \(X\) be a topological space. A path from point \(a\) to point \(b\) in \(X\) is a continuous function \(p: [0,1] \to X\) such that \(p(0) = a\) and \(p(1)=b\text{.}\)
Section Introduction
We defined connectedness in terms of separability by open sets. There are other ways to look at connectedness. For example, the subset \((0,1)\) is connected in \(\R\) because we can draw a line segment (which we will call a path) between any two points in \((0,1)\) and remain in the set \((0,1)\text{.}\) So we might alternatively consider a topological space to be connected if there is always a path from one point in the spaced to another. Although this is a new notion of connectedness, we will see that path connectedness and connectedness are related.
Intuitively, a space is path connected if there is a path in the space between any two points in the space. To formalize this idea, we need to define what we mean by a path. Simply put, a path is a continuous curve between two points. We can therefore define a path as a continuous function.
With the notion of path, we can now define path connectedness.
Definition 19.2.
A subspace \(A\) of a topological space \(X\) is path connected if, given any \(a, b \in A\) there is a path in \(A\) from \(a\) to \(b\text{.}\)
Preview Activity 19.1.
(a)
Is \(\R\) with the Euclidean metric topology path connected? Explain.
(b)
Is \(\R\) with the finite complement topology path connected? Explain.
(c)
Let \(A = \{b,c\}\) in \((X, \tau)\) with \(X= \{a,b,c,d,e,f\}\) and
\begin{equation*}
\tau = \{\emptyset,\{a\}, \{c,d\}, \{a,c,d\}, \{b,c,d,e,f\}, X\}\text{.}
\end{equation*}
Is \(A\) connected? Is \(A\) path connected? Explain.
