Example: Let "X" be the set of natural numbers {0, 1, 2, ...} and let "Y" be the set {0} ∪ {1, 1/2, 1/3, ...}, both with the subspace topology from the real line. Define "f": "X" → "Y" by mapping 0 to 0 and "n" to 1/"n" for positive integers "n". Then "f" is continuous, and in fact a weak homotopy equivalence, but it is not a homotopy equivalence. Thus the naive homotopy category distinguishes spaces such as "X" and "Y", whereas they become isomorphic in the homotopy category.