Juju Saludos 😊🤩

Homotopy Equivalent of X and Y

Let X and Y be topological Spaces

f: X ---> Y
g: Y ---> X

such that f o g ~ id_Y
and g o f ~ id_X

Denote that homotopy equivalence is weaker than homeomorphism.
It's hard to study Homotopy equivalence by itself, because we need to come up with two continuous maps and two homotopies rather, Algebraic Topology focuses on Invariants of spaces, leading to homotopy invariants. For this Homology is a nice property that is transferred functorially to other homotopy equivalent Spaces.

Disculpa, vendes tu cuenta?

Clarifications:

Vector spaces are usually over R or C.
Vector bundles need to be 'locally trivial', whatever that means.
Whatever that means has a meaning! It means that the fibres are mapped "nicely" without any "twist". For instance let E be the total space, B the base space and F the fiber. When there is "Trivial fiber bundle", π: E = BxF ---> B happens, requiring at least surjectivity.

Some counter examples:

Mobius Strip
Klein Bottle

Vector Bundle: It's an attempt to form a precise definition of a family of vector spaces 'parameterized' by another space (topological, manifold or a variety) for every x in X there is a V(x) in VB