Local path connectedness is a topological property of a space where every point has a neighborhood that is path connected, meaning that there is a path connecting any two points within that neighborhood.
Path connectedness refers to the entire space being connected by a path, while local path connectedness only requires each point to have a path connected neighborhood. This means that a space can be path connected without being locally path connected.
Local path connectedness is important because it helps to classify and distinguish different types of topological spaces. It also has applications in mathematical analysis and differential geometry.
Yes, a space can be locally path connected but not path connected. For example, the topologist's sine curve is locally path connected, but not path connected.
To prove that a space is locally path connected, you can use the definition of local path connectedness and show that every point in the space has a path connected neighborhood. Alternatively, you can use the fact that a space is locally path connected if and only if it is locally connected and semi-locally simply connected.