In topology, contractibility refers to a property of a space that can be "contracted" continuously to a point. In other words, a space is contractible if every point in the space can be continuously deformed to a single fixed point within the space. Formally, let X be a topological space and x0 be a fixed point in X. X is said to be contractible if there exists a continuous map F: X × [0,1] -> X such that F(x,t) = x for all x in X and F(x0,t) = x0 for all t in [0,1]. This means that for any point x in X, we can continuously deform it to the point x0 using the map F. Contractibility is an important concept in algebraic topology because it provides a way to understand the structure of spaces. Contractible spaces are often used as building blocks or reference spaces when studying more complicated topological structures. For example, many results and techniques in algebraic topology are based on the fact that contractible spaces have trivial homotopy groups (a measure of how loops behave in a space). A simple example of a contractible space is Euclidean n-space (R^n), where any point can be continuously moved to the origin by scaling. Similarly, any convex subset of R^n is also contractible. On the other hand, spaces with holes or nontrivial topological features are typically not contractible. Contractibility is closely related to homotopy equivalence, which is a stronger notion of similarity between spaces. If two spaces are homotopy equivalent, then they are also contractible. However, not all contractible spaces are homotopy equivalent. Overall, contractibility provides a fundamental concept for understanding and classifying topological spaces based on their ability to deform continuously to simpler forms. # References ```dataview Table title as Title, authors as Authors where contains(subject, "contractibility") ```