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A continuous transformation of one function or shape into another that can be performed without tearing or gluing.

Denoted as $\pi_1(X, x_0)$, it represents the set of all loops at a point $x_0$ in space $X$ up to homotopy.

A mathematical concept that provides a way to algebraically classify topological spaces according to the number and type of holes of different dimensions.

A space is simply connected if it is path-connected and every loop can be continuously tightened to a point.

A subspace to which a space can be continuously contracted without changing its topological nature.

A topological space that maps onto another space (the base space) such that locally around every point in the base space, the mapping resembles a product of the local topology with a discrete set of points.

A type of equivalence between topological spaces that exists if two spaces can be transformed into one another through a homotopy.

Groups that deal with classes of maps from spheres of higher dimensions into a space, generalizing the concept of the fundamental group.

The classification of a topological space based on its structure under homotopy equivalence.

A particular kind of mapping between topological spaces that has properties making it amenable to analysis via homotopy theory.

A type of topological space that is a key object of study in algebraic topology and homotopy theory, characterized by having a single nontrivial homotopy group.