prompter: explain the definition of connectedness in topology, and explain the intuition

assistant: 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.

prompter: Can you give an example of a connected space that is not simply connected?

assistant: 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.

assistant: For example, neither a doughnut nor a coffee cup (with a handle) is simply connected, but a hollow rubber ball is simply connected. In two dimensions, a circle is not simply connected, but a disk and a line are.

assistant: I can give an example, however, as a language model, I cannot produce a graphic. Imaging a social media platform where people can add each other as friends. In this example, we will have 5 users, A,B,C,D, and E. 
Let's say that each of the 2 adjacent letters [A,B],[B,C],[C,D],[D,E] are friends. In this case, since any user can follow a "path" to any other user, the network is connected. 
If instead [C,D] were not friends, then A,B,C could not reach D and E, so it would not be connected. 
Additionally, if every user is friends with every other user, we can call it 'Complete'

assistant: Topologically, a connected space is one that cannot be represented as the union of two or more disjoint and non-empty spaces. In other words, a connected space is not made up of multiple separate spaces. Intuitively, it can be thought of as being "in one piece," with the term referring to the fact that all the points within the space are connected to each other.

assistant: Connectedness in topology refers to the notion that a space is a single connected piece, rather than being composed of multiple pieces. Intuitively, this means that if you draw a line in the space, you can draw it so that it doesn't leave any gaps or holes, and it passes through every point in the space. Formally, connectedness is defined as a space that cannot be written as the union of two disjoint non-empty open sets. This means that if you take two non-empty open sets in the space, they cannot have no points in common, and therefore cannot be separated by a line. This definition captures the intuition that a space is either connected or not, rather than having varying degrees of connectedness.

prompter: What does "disjoint" mean? Could you go a bit more into detail in a formal way?

prompter: Can you please make it easier to understand?

assistant: Roughly speaking, a connected topological space is one that is “in one piece”. The way we
will define this is by giving a very concrete notion of what it means for a space to be “in two or
more pieces”, and then say a space is connected when this is not the case.

assistant: Imagine there are 0 ways to get from set A to set B. In this case, it is disjoint. If there was 1 way, it would be connected. As long as there is 1 way to get from a given location to any other given location, it remains connected.

assistant: A space is connected when there is at least a way to go from one point of it (origin) to another point of it (destination) without having to pass through any point that doesn't belong to this space. Let's say a space is connected when it's like one unique island, regardless of its shape.