I know it’s a basic question but topology terminology is hard for me
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An *isomorphism* is a (usually) reversible, structure-preserving transformation. Which kind of structures are preserved determines what kind of isomorphism it is. If the transformation preserves the topological structure of an object, it’s called a *homeomorphism*. If it preserves the algebraic structure of the object, it’s a *homomorphism*. If it preserves the object’s sense of distance, it’s an *isometry*. And so on.
In essence, a homeomorphism is a specific kind of isomorphism.
Both are one-to-one (injective and surjective), and as such have an inverse function.
Homeomorphisms are not only one-to-one but also preserve the topological structure. For that the function has to be continuous and its inverse too. So from a strictly topological point of view if a space/manifold/whatever A is homeomorphic to space/manifold/whatever B, they can be considered identical.
Isomorphisms are not only one-to-one but also preserve the algebraic structure. For that the function has to be compatible with the algebraic operations, and its inverse too. So for example if a group A is isomorphic (as a group) to a group B, they can be considered identical. And the isomorphism (let’s call it f) has to play nicely with the group operations: f(x*y) = f(x).f(y), where * is the operation in A and . the operation in B. And if your groups also have topologies, you’d be interested in isomorphisms which are also homeomorphisms.
There is a similar concept for differentiable manifolds, called diffeomorphism.
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