Extension of a topological group
Imagine you have a group of friends who all like playing together, but some of them can only play at certain times or in certain places. You wish you could find a way to make sure everyone can come and play whenever they are available. This is a bit like an extension of a topological group.
A topological group is a group of friends who not only like playing together but also have a special way of hugging each other, which they do every time they meet up. This hug gives them a sense of closeness and comfort.
Now imagine that some of your friends live too far away from each other to hug every time they meet up, or they want to play at different times or places. To solve this problem, you might want to create an extension of your group, which would be a new group that includes all of your friends, regardless of where they are or when they want to play.
The new group would need to have the same special hug as the original group, which means that it would be a topological group as well. The two groups would be like siblings – they share the same special hug, but they can play together or separately.
This is what an extension of a topological group is – a new group that includes the original topological group and additional elements that satisfy the same special hug rule. Instead of playing games, members of the group might perform mathematical operations like addition or multiplication, but they still need to follow the same rules of closeness and comfort that make them a topological group.
A topological group is a group of friends who not only like playing together but also have a special way of hugging each other, which they do every time they meet up. This hug gives them a sense of closeness and comfort.
Now imagine that some of your friends live too far away from each other to hug every time they meet up, or they want to play at different times or places. To solve this problem, you might want to create an extension of your group, which would be a new group that includes all of your friends, regardless of where they are or when they want to play.
The new group would need to have the same special hug as the original group, which means that it would be a topological group as well. The two groups would be like siblings – they share the same special hug, but they can play together or separately.
This is what an extension of a topological group is – a new group that includes the original topological group and additional elements that satisfy the same special hug rule. Instead of playing games, members of the group might perform mathematical operations like addition or multiplication, but they still need to follow the same rules of closeness and comfort that make them a topological group.