Projective topology

Alright, kiddo! Let's talk about projective topology.

Imagine you have some shapes, like circles or triangles or squares. Projective topology is a way of looking at these shapes and understanding how they are related to each other.

But first, we need to talk about what we mean by "related." Have you ever played with a toy that has different pieces that fit together? Like maybe you have a puzzle with different shapes that fit into a board? That's kind of like what we mean by related in projective topology. We're interested in how different shapes can fit together in different ways.

Now, let's talk about the word "projective." Have you ever made a shadow puppet with your hands? When you make a shadow puppet, you project an image of your hand onto the wall. Projective topology is kind of like that, but instead of shadows, we're thinking about how shapes look when we shine a light on them.

So, let's say you have a circle and a square. If you shine a light on them from some angles, they'll look the same size and shape. But if you move the light to a different angle, they might look different. In fact, if you shine the light from just the right angle, the circle and the square might look like they have the same shape! That's because projective topology is interested in how shapes look when we change the perspective we're looking at them from.

One way to think about projective topology is to imagine looking at a 3D shape from different angles. If you have a 3D cube, it will look like a square from some angles, but from other angles it will look like a rectangle or a trapezoid. Projective topology is about understanding how these different perspectives are related to each other.

So, in summary, projective topology is a fancy way of thinking about how different shapes can fit together and look the same or different from different angles. It's like playing with puzzles or making shadow puppets, but with more math involved!