Homotopy lifting property

In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function from a topological space E to another one, B. It is designed to support the picture of E "above" B by allowing a homotopy taking place in B to be moved "upstairs" to E.

For example, a covering map has a property of unique local lifting of paths to a given sheet; the uniqueness is because the fibers of a covering map are discrete spaces. The homotopy lifting property will hold in many situations, such as the projection in a vector bundle, fiber bundle or fibration, where there need be no unique way of lifting.

Assume from now on all mappings are continuous functions from a topological space to another. Given a map ${\displaystyle \pi \colon E\to B}$, and a space ${\displaystyle X\,}$, one says that ${\displaystyle (X,\pi )\,}$ has the homotopy lifting property, or that ${\displaystyle \pi \,}$ has the homotopy lifting property with respect to ${\displaystyle X\,}$, if:

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