Definition: Let f: X → Y and g: X → Y be continuous maps between topological spaces X and Y . The maps f and g are said to be homotopic if there exists a continuous map                    H: X × [0, 1] → Y such that H(x, 0) = f(x)  and H(x, 1) = g(x) for all x ∈ X. If the maps f and g are homotopic then we denote this fact by writing f ≃g. The map H with the properties stated above is referred to as a homotopy between f and g.

Continuous maps f and g from X to Y are homotopic if and only if it is possible to ‘continuously deform’ the map f into the map g.

Lemma 1.1 Let X and Y be topological spaces. The homotopy relation ≃ is an equivalence relation on the set of all continuous maps from X to Y .

Proof Clearly f≃f, since (x, t) → f(x) is a homotopy between f and itself. Thus the relation is reflexive. If f ≃ g then there exists a homotopy H: X × [0, 1] → Y between f and g (so that H(x, 0) = f(x) and H(x, 1) = g(x) for all x ∈ X). But then (x, t) → H(x, 1 − t) is a homotopy between g and f. Therefore f ≃ g if and only if g ≃ f. Thus the relation is symmetric. Finally, suppose that f ≃g and g ≃h. Then there exist homotopies H₁: X × [0, 1] → Y and H₂: X × [0, 1] → Y such that H₁(x, 0) = f(x), H₁(x, 1) = g(x) = H₂(x, 0) and H₂(x, 1) = h(x) for all x ∈ X. Define H: X × [0, 1] → Y by

Now H|X ×[0, 1/2] and H|X ×[1/2 , 1] are continuous. It follows from elementary point set topology that H is continuous on X × [0, 1]. Moreover H(x, 0) = f(x) and H(x, 1) = h(x) for all x ∈ X. Thus f ≃ h. Thus the relation is transitive. The relation ≃ is therefore an equivalence relation.

Definition: Let X and Y be topological spaces, and let A be a subset of X.

Let f: X → Y and g: X → Y be continuous maps from X to some topological space Y , where f|A = g|A (i.e., f(a) = g(a) for all a ∈ A). We say that f and g are homotopic relative to A (denoted by f ≃g rel A) if and only if there exists a (continuous) homotopy H: X × [0, 1] → Y such that H(x, 0) = f(x)  and H(x, 1) = g(x) for all x ∈ X and H(a, t) = f(a) = g(a) for all a ∈ A.