Disjoint union (topology)
Imagine you have two different toys, a ball and a teddy bear, and you want to put them together in one toy box. However, you don't want them to mix with each other, and you want to be able to tell which toy is which just by looking at them.
This is similar to what happens in topology when we have two different sets, A and B, and we want to combine them into a single set, say C. The disjoint union of A and B, denoted A∐B, is a new set that contains all the elements of both A and B, but keeps them separate from each other.
In other words, when we take the disjoint union of A and B, we create a new set C that has all the elements of A and B, but none of them are repeated or mixed up. We can tell which elements come from A and which ones come from B just by looking at them.
To construct the disjoint union, we create two copies of A and B, and label each element in the copy of A with the letter "a" and each element in the copy of B with the letter "b". Then, we put the two sets together in a single set C, so that the elements of A are all labeled with "a" and the elements of B are all labeled with "b". Because the labeling keeps the elements distinct, we can form C as the union of the labeled copies of A and B.
For example, suppose A = {1,2,3} and B = {a,b}. Then, the disjoint union of A and B is A∐B = {(1,a),(2,a),(3,a),(a,b),(b,b)}.
The disjoint union is an important concept in topology because it allows us to combine different topological spaces in a way that preserves their separate identities. For example, if we have two spaces X and Y, we can take their disjoint union X∐Y to create a new space that contains X and Y as subspaces, but otherwise keeps them separate from each other. This can be useful for constructing more complicated spaces, or for studying the properties of spaces that have multiple components.
This is similar to what happens in topology when we have two different sets, A and B, and we want to combine them into a single set, say C. The disjoint union of A and B, denoted A∐B, is a new set that contains all the elements of both A and B, but keeps them separate from each other.
In other words, when we take the disjoint union of A and B, we create a new set C that has all the elements of A and B, but none of them are repeated or mixed up. We can tell which elements come from A and which ones come from B just by looking at them.
To construct the disjoint union, we create two copies of A and B, and label each element in the copy of A with the letter "a" and each element in the copy of B with the letter "b". Then, we put the two sets together in a single set C, so that the elements of A are all labeled with "a" and the elements of B are all labeled with "b". Because the labeling keeps the elements distinct, we can form C as the union of the labeled copies of A and B.
For example, suppose A = {1,2,3} and B = {a,b}. Then, the disjoint union of A and B is A∐B = {(1,a),(2,a),(3,a),(a,b),(b,b)}.
The disjoint union is an important concept in topology because it allows us to combine different topological spaces in a way that preserves their separate identities. For example, if we have two spaces X and Y, we can take their disjoint union X∐Y to create a new space that contains X and Y as subspaces, but otherwise keeps them separate from each other. This can be useful for constructing more complicated spaces, or for studying the properties of spaces that have multiple components.
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