# Sudden contracting homotopy

This term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it.

This article defines a property of a homotopy from a topological space to itself

## Definition

Let $X$ be a topological space. A sudden contracting homotopy on $X$ is a map $F: X \times I \to X$ such that:

• $F(x,0) = x \ \forall \ x \in X$
• For any $t < 1$, the map sending $x$ to $F(x,t)$ is a homeomorphism
• There exists a point $x_0 \in X$ such that $F(x,1) = x_0 \ \forall \ x \in X$

A topological space which has a sudden contracting homotopy is termed a suddenly contractible space.