Universal spaces in the topology and topological dynamics

Universal spaces in topology are like the biggest playgrounds you can imagine. Just like how you have a favorite playground that has all the cool stuff like swings, slides, monkey bars, and a sandpit, universal spaces have all the cool topological properties that make them great to study.

In topology, we are interested in studying spaces and how they can be connected or how they can be pulled or stretched to become similar to other spaces. Universal spaces are like the biggest playground where you can play any topological game you want. They are very special because they have the ability to contain copies of any other space. So if you want to study a specific space, you can go to the universal space and find a copy of it there.

But just like how you can't play in every playground if you don't know how to get there, you might need some special instructions to get to these universal spaces. That's where topological dynamics comes in.

Topology and dynamics often go hand in hand because topology studies how spaces behave, while dynamics studies how things move or change over time. In topological dynamics, we study how spaces change over time, especially how they change under continuous transformations. We look at things like orbits, fixed points, and attractors to see how the space behaves under certain transformations. This can help us understand how we can get to different universal spaces and even create new ones.

So, in summary, universal spaces in topology are like super cool playgrounds that have copies of every other space. Topological dynamics helps us understand how these spaces can be reached and how we can study their topological properties.