Two end points of a path can be connected. You can use the Knife tool cut to split a path.
How to connect two end points
    1. Select the Connect tool connect from the Tools panel.
    2. Click one point that you want to connect to the next.
    3. Click the second point.
    
            A path is created between the two points.
How to join two end points that are within each others snap distance
    1. Select the Join tool join from the Tools panel.
    2. Drag a selection frame with the Direct Selection tool and capture two points that are within each others snap distance.
        –OR–
        Select and drag a point to another point.
The two points are joined if they are within each others snap distance.
How to split a path
    1. Select a path.
    2. Select the Split tool cut from the Tools panel.
    3. Drag a line over a path and release the mouse button.
    two-points
        An anchor point is shown at the intersection and the resulting new paths can now be selected separately.
How to create a mask
An extra function of the Split tool is to create masks. Using the CTRL key (Windows) or Command key (MacOS), you can cut out parts of an object. This single click action is roughly the same as making a compound of two paths and excluding the shape of the top object from the bottom object. Please note that the objects must be paths.
How to split two objects from each other
    1. Select two objects that are placed on top of each other.
    2. If the selected objects are no paths, choose Extra > Convert to Path.
    3. Select the Split tool cut from the Tools panel.
    4. With all objects selected, click the top path (you can click a point or the path).
    5. Use the Selection tool to move the top object away from the underlying object.
   cut-shape
 cutshape2