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ComputerGeek
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What is the difference?
Don't you mean "homeomorphic vs isotropic"?ComputerGeek said:What is the difference?
So, it is appropriate to say:hypermorphism said:An isotopy is a smooth path of embeddings between two manifolds, while a homeomorphism is just a single function between two manifolds. Ie., a right circular cylinder centered at the origin with unit radius is a representation of an isotopy between the two circles at either end.
While the unlink of 2 components is homeomorphic to the Hopf link, the two are not isotopic.
Homeomorphic and isotopic are two terms used in topology, the branch of mathematics that studies the properties of geometric objects that remain unchanged under continuous deformations. Homeomorphic refers to two objects that can be transformed into each other by stretching, bending, and twisting without tearing or gluing. Isotopic, on the other hand, refers to two objects that can be transformed into each other by continuously deforming one into the other, without any cuts, punctures, or self-intersections.
Yes, two homeomorphic objects can also be isotopic. This means that if two objects can be transformed into each other without tearing, then they can also be deformed into each other without any cuts or punctures. In other words, homeomorphic implies isotopic, but the converse is not always true.
A classic example is a donut and a coffee mug. Both objects have only one hole, and can be transformed into each other by stretching or compressing, making them homeomorphic. However, a donut cannot be transformed into a coffee mug without making a hole in the donut or cutting the mug, so they are not isotopic.
Homeomorphic and isotopic are properties of the objects themselves, not their representations. This means that even if two objects have different shapes or appearances, they can still be homeomorphic or isotopic if they can be transformed into each other by continuous deformations.
Homeomorphism and isotopy are useful in various scientific fields, including biology, physics, and chemistry. In biology, these concepts are used to study the shapes and structures of organisms, such as understanding how different species are related through their evolutionary history. In physics, they are used to describe the properties of space and objects, such as the behavior of particles or the topology of the universe. In chemistry, they are used to predict and understand the properties of molecules and their reactions. Overall, homeomorphism and isotopy help scientists to better understand and classify the complex structures and phenomena in our world.