Connected vs. path connected

A topological space $X$ is said to be connected if it cannot be represented as the union of two disjoint, nonempty, open sets. While this definition is rather elegant and general, if $X$ is connected, it does not imply that a path exists between any pair of points in $X$ thanks to crazy examples like the topologist's sine curve:

$$X = \{ (x,y) \in {\mathbb{R}}^2 \;\vert\; x = 0 y = \sin(1/x) \} .$$ (4.8)

Consider plotting $X$. The $\sin(1/x)$ part creates oscillations near the $y$-axis in which the frequency tends to infinity. After union is taken with the $y$-axis, this space is connected, but there is no path that reaches the $y$-axis from the sine curve.

How can we avoid such problems? The standard way to fix this is to use the path definition directly in the definition of connectedness. A topological space $X$ is said to be path connected if for all $x_1,x_2 \in X$, there exists a path $\tau$ such that $\tau(0) = x_1$ and $\tau(1) = x_2$. It can be shown that if $X$ is path connected, then it is also connected in the sense defined previously.

Another way to fix it is to make restrictions on the kinds of topological spaces that will be considered. This approach will be taken here by assuming that all topological spaces are manifolds. In this case, no strange things like (4.8) can happen,^4.7 and the definitions of connected and path connected coincide [451]. Therefore, we will just say a space is connected. However, it is important to remember that this definition of connected is sometimes inadequate, and one should really say that $X$ is path connected.