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If X is a space, the deleted product space, X*, is Xx XD, where D is the diagonal. If Y is a space and f is a continuous map from X to Y, then Xf/ is the inverse image of Y* under the map f x f taking Xx X into Yx Y. In this paper, we investigate the following questions: "What maps f are such that Xf* is homotopically equivalent to X* ", and "What maps f are such that Xf* is homotopically equivalent to f(X)* ?" If X is the union of two nondisjoint simplexes and f is a simplicial map from Xx X such that f lf(X) is one-to-one, we obtain necessary and sufficient conditions for Xf* and f(X)* to be homotopically equivalent. If X is the union of nondisjoint simplexes A and B with dim B= 1+ dim (A r) B), we obtain necessary and sufficient conditions for X* and Xf to be homotopically equivalent if f is in the class of maps mentioned.

Deleted products of spaces which are unions of two simplexes