Topological space

We use a cookbook to illustrate the idea of a topological space, Little One. Do you remember when we baked cookies last week? Before we could place the cookies in the oven, we had to cut the dough into cookie shapes using our cookie cutter. This cookie cutter helped us determine the shape of our cookies.

In the same way, a topological space involves a set of points that are organized in a particular way, using something similar to a cookie cutter. Except we don't use a cookie cutter; we use what is called a "topology." A topology tells us which points are close to each other, and which points are far apart.

Let's say we have a set of four points, which we will call A, B, C, and D. We could arrange these points in different ways, like lining them up in a row or placing them in a square formation. In a topological space, we want to be able to tell how near or far apart each point is from the others.

So, let's say that in our topological space, we say that A and B are close together because they are next to each other in a row, but C and D are farther apart because they are diagonally across from each other. This tells us how we can travel from one point to another, and which paths we can take.

Now, here's where things get tricky. Sometimes, we can have a set of points that are close together in one topology, but farther apart in another. For example, if we move the points A, B, C, and D around, we might be able to make it so that A is close to C and far from B, whereas before, A was close to B and far from C.

This is why we call this a "topological space." It's kind of like a cookie space, where we don't know what the cookies will look like until we use our cookie cutter, or topology, to determine their shape. This helps us understand how the points are related to each other, and which paths we can take to move between them.