Direct sum of topological groups

Okay kiddo, so you know what a group is, right? It's a bunch of things that you can put together in a certain way, like adding or multiplying. And a topological group is just like a regular group, but it also has a special way of being put together, called a "topology".

Now, imagine you have two topological groups, let's call them G and H. A direct sum of these two groups is when you take all the elements in G and all the elements in H, and you put them together to create a new group in a specific way.

Here's how you do it: first, you take an element from G and an element from H. You put these two elements together in a special way, which is just like stacking them on top of each other. So if you had an element "a" from G and an element "b" from H, you would stack them like this: (a, b).

This new element, (a, b), is now part of the direct sum group. And you do this for every combination of elements in G and H. So if G had three elements and H had four elements, you would have twelve new elements in the direct sum group.

But wait, there's more! Remember how we said that a topological group has a special way of being put together? Well, the direct sum of two topological groups also has a special topology. This means that the elements in the direct sum group have a certain structure and can be arranged in a special way.

For example, let's say that G and H were both circles (like a donut shape). When you take the direct sum of these two groups, you would get a set of pairs that look like two circles stacked on top of each other. This might look like a cylinder shape. And the special topology of this direct sum group means that you can slide this cylinder shape around and still have the same structure.

So in short, the direct sum of two topological groups is when you stack all the elements together in a certain way to make a new group, and this new group has a special shape that you can move around in a specific way.