Eilenberg–Steenrod axioms
Imagine you have a magician's hat, and inside the hat, there are different colored balls. Each ball represents a shape or an object in mathematics. Now, think about how you might want to study and understand these shapes and objects.
One way you can study them is by picking out one ball at a time and looking at its properties and characteristics. For example, you can look at a ball that represents a triangle and see how many sides it has, what its angles are, and how you can draw it on a piece of paper. These are called "local" properties because you are only considering one shape at a time.
But what if you want to understand how these shapes relate to each other? How they can be transformed into one another or combined to create something new? This is where the Eilenberg-Steenrod axioms come into play.
The Eilenberg-Steenrod axioms provide a set of rules or guidelines for studying how shapes can be connected or transformed. They help us understand the "global" properties of shapes, which means looking at the big picture and how different shapes fit together.
One of the axioms says that there should be a way to connect two shapes together to create a new shape. Think of this as taking two balls, say a triangle and a square, and sticking them together to form a new shape called a pentagon. This rule helps us understand how shapes can be combined and transformed into something new.
Another axiom says that there should be a way to transform a shape without changing its essential properties. This is like when you take a ball that represents a square and you can rotate it or flip it over without changing the fact that it's still a square. This rule helps us understand how shapes can be changed but still keep their important characteristics.
There are other axioms as well, like one that says you can take a shape and cut it into smaller pieces, or one that says you can take a shape and stretch or shrink it without changing its essential properties.
By following these rules, mathematicians can study shapes and objects in a more systematic and organized way. They can understand how shapes can be connected and transformed, and how they relate to each other. This is really important because it helps us understand the fundamental building blocks of mathematics and how different mathematical concepts are connected.
One way you can study them is by picking out one ball at a time and looking at its properties and characteristics. For example, you can look at a ball that represents a triangle and see how many sides it has, what its angles are, and how you can draw it on a piece of paper. These are called "local" properties because you are only considering one shape at a time.
But what if you want to understand how these shapes relate to each other? How they can be transformed into one another or combined to create something new? This is where the Eilenberg-Steenrod axioms come into play.
The Eilenberg-Steenrod axioms provide a set of rules or guidelines for studying how shapes can be connected or transformed. They help us understand the "global" properties of shapes, which means looking at the big picture and how different shapes fit together.
One of the axioms says that there should be a way to connect two shapes together to create a new shape. Think of this as taking two balls, say a triangle and a square, and sticking them together to form a new shape called a pentagon. This rule helps us understand how shapes can be combined and transformed into something new.
Another axiom says that there should be a way to transform a shape without changing its essential properties. This is like when you take a ball that represents a square and you can rotate it or flip it over without changing the fact that it's still a square. This rule helps us understand how shapes can be changed but still keep their important characteristics.
There are other axioms as well, like one that says you can take a shape and cut it into smaller pieces, or one that says you can take a shape and stretch or shrink it without changing its essential properties.
By following these rules, mathematicians can study shapes and objects in a more systematic and organized way. They can understand how shapes can be connected and transformed, and how they relate to each other. This is really important because it helps us understand the fundamental building blocks of mathematics and how different mathematical concepts are connected.