Redirected from Homotopy equivalent
Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function H : X × [0,1] → Y from the product of the space X with the unit interval [0,1] to Y such that, for all points x in X, H(x,0)=f(x) and H(x,1)=g(x).
Being homotopic is an equivalence relation on the set of all continuous functions from X to Y. This homotopy relation is compatible with function composition in the following sense: if f_{1}, g_{1} : X → Y are homotopic, and f_{2}, g_{2} : Y → Z are homotopic, then their compositions f_{2} o f_{1} and g_{2} o g_{1} : X → Z are homotopic as well.
This allows to define the homotopy category: the objects are topological spaces, and the morphisms are homotopy classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy equivalent in the following sense: there exist continuous maps f : X → Y and g : Y → X such that g o f is homotopic to the identity map id_{X} and f o g is homotopic to id_{Y}. The maps f and g are called homotopy equivalences in this case.
Intuitively, two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. For example, a solid disk or solid ball is homotopy equivalent to a point, and R^{2}  {(0,0)} is homotopy equivalent to the unit circle S^{1}. Those spaces that are homotopy equivalent to a point are called contractible.
Homotopy equivalence is important because in algebraic topology most concepts cannot distinguish homotopy equivalent spaces: if X and Y are homotopy equivalent, then
Especially in order to define the fundamental group, one needs the notion of homotopy relative to a subspace. These are homotopies which keep the elements of the subspace fixed. Formally: if f and g are continuous maps from X to Y and K is a subset of X, then we say that f and g are homotopic relative K if there exists a homotopy H : X × [0,1] → Y between f and g such that H(k,t) = f(k) for all k∈K and t∈[0,1].
In case the two given continuous functions f and g from the topological space X to the topological space Y are homeomorphisms, one can ask whether they can be connected 'through homeomorphisms'. This gives rise to the concept of isotopy, which is a homotopy H in the notation used before, such that for each fixed t, H(x,t) gives a homeomorphism.
In geometric topology  for example in knot theory  the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same? We take two knots K_{1} and K_{2} in threedimensional space. The intuitive idea of deforming one to the other should correspond to a path of homeomorphisms: an isotopy starting with the identity homeomorphism of threedimensional space, and ending at a homeomorphism h such that h moves K_{1} to K_{2}.
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