Topology: An Inquiry-Based Approach

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 $\mathbb{R}$ 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.