explain the definition of connectedness in topology, and explain the intuition
In mathematics, connectedness is a property of topological spaces that describes how points within a space are related to one another. A topological space is said to be connected if it is not the union of two or more disjoint non-empty open sets.

The intuition behind connectedness is to think of a space as a network of points, where each point is connected to some other points within the space. Connectedness then says that, for any two points in the space, there exists a path connecting the two points that lies entirely within the space. In other words, there are no islands or holes in the space, and every point can be reached from every other point by traveling along paths that lie within the space.

For example, consider the real line, which is a connected topological space. This means that given any two points on the real line, there exists a path connecting the two points that lies entirely on the real line. On the other hand, the plane with the origin removed is not connected, because it can be divided into two separate connected components: one consisting of the points with positive x-coordinates, and the other consisting of the points with negative x-coordinates.

Connectedness is an important property in topology, as it gives us a way to determine whether two points in a space are close to one another or not, and it is also closely related to other topological concepts such as continuity and connected components.
Can you give an example of a connected space that is not simply connected?
Sure. The torus is an example of a connected space that is not simply connected. A torus is a two-dimensional surface that is shaped like a doughnut, and it has one hole running through it. This means that although it is connected, it is not simply connected. This is because any two points on the surface of the torus can be connected by a path that lies entirely on the surface, but this path cannot be shrunk down to a single point without passing through the hole.