Homotopy
Relative homotopy

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 to K if there exists a homotopy H : X × → Y between f and g such that H (k, t) = f (k) = g (k) for all k ∈ K and t ∈. Also, if g is a retraction from X to K and f is the identity map, this is known as a strong deformation retract of X to K.When K is a point, the term pointed homotopy is used.
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