Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A continuous deformation of one continuous function to another.
 n. A theory associating a system of groups to each topological space.
 n. A system of groups associated to a topological space.
from The Century Dictionary and Cyclopedia
 n. The development of an embryo into an adult organism by a series of changes which occur in the parts of the body in which they occurred in the development of its parents.
Etymologies
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Examples

Some details must be ironed out: for example, the investigation of loops which are very similar gives no insight into the structure of the space, so we work instead with sets of loops called homotopy classes One of the most useful tools in algebraic topology is the [[fundamental group]],

(humorously) a dunce cap, its fundamental group is its set of path loops that has a useful characteristic known as "[[homotopy]]"

However, I find to stomach arguments that other completely useless things – like calculating the QCD beta function to n loops, or calculating the homotopy groups of spheres is somehow more worthy.

So I presume that the cyclic group, or homotopy on the 3sphere persists for Nlarge or for classical recovery.

On the typical serious math paper, say, noncommutative loops over Lie algebra, or real homotopy theory of Kaehler manifolds, or compactification on CalabiYau manifolds, my head just starts to spin.

See: on homotopy, where the space Y is in fact euclidean 3space.

I believe that any confusion here originates from the use of homotopy in knot theory where an ignorance of the space in which the knot is embedded would result in every knot being equivalent to the trivial knot.

Welcome to the wonderful world of homotopy equivalence classes.

This book will be of interest to both topologists and algebraists, particularly those concerned with homotopy theory.

These gadgets show up in modern approaches to universal algebra like questions, and also all over the place in topology; some of the earliest instances were from homotopy theory.
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