General topology

Topology is like a super-flexible way to look at shapes. Think about a rubber band. You can stretch it, squish it, or bend it, and it's still a rubber band.

In topology, we care more about how a shape is connected than its exact size or straightness. A donut and a coffee mug are actually the same in topology because you can change one into the other without ripping or gluing. It's all about the holes!

This game of shapes started a long, long time ago with smart people like Leonhard Euler. He looked at a puzzle about walking across seven bridges in a city called Königsberg without crossing any bridge twice. He figured out it was impossible!

Later, mathematicians like Bernhard Riemann and Felix Hausdorff explored these ideas more, creating the rules for what we now call topology. It's like building a whole new playground for shapes!

Topology helps scientists understand all sorts of things. Imagine trying to design a new video game character or figure out how a tiny virus is shaped. Topology gives them the tools to think about shapes in a special way. It's also used in amazing places like understanding how the universe is put together or even how our brains work. It’s a secret language for understanding the world around us!

In topology, we call shapes 'topological spaces'. The main idea is something called 'open sets'. Think of them like little neighborhoods on a map.

If you can find a tiny open set around any point in your shape, then the shape is a topological space. It's like making sure every house in a neighborhood has a path to the main road. This helps us understand how shapes are connected and how they behave when stretched.