Close maps are homotopic
Suppose is a topological space and is a compact subset in Euclidean space, such that there exists an open subset such that is a strong deformation retract of . Then, there exists such that if two maps are -close, in the sense:
(where denotes the Euclidean distance) then and are homotopic.
A concrete interpretation of this is as follows. Suppose we view as a compact metric space with the metric induced as a subset of . Then we can give the topology of uniform convergence. There is a natural map:
where denotes the space of homotopy classes of continuous maps from to . The above result says that the above map is continuous if we give the discrete topology. This interpretation follows because for every function in a homotopy class, the -neighbourhood of that function is also in the same homotopy class.
The advantage of this interpretation is that for a compact metric space, the topology of uniform convergence coincides with the compact-open topology, which can be defined without reference to the explicit metric. Thus, we can state the result more abstractly as: