Consideration of the space of loops on a given space has played an important role in algebraic topology since Serre generalized the notion of fibre space and applied spectral homology technique to the computation of homotopy groups. However, loop-spaces have the disadvantage of being very ``large'' and not susceptible to the familiar combinatorial techniques. In this paper the author removes this disadvantage, at least under fairly general conditions. More precisely, let \(A\) be a countable \(CW\)-complex with a single vertex, \(a^0\), the base point. Such a complex is called special; and the author constructs a special complex \(A_{\infty}\) and a map \(\alpha\) of \(A_{\infty}\) into the space, \(\Omega\), of loops on the suspension of \(A\) which induces homology and homotopy isomorphisms. Moreover, \(A\) is embedded in \(A_{\infty}\) and in \(\Omega\) and \(\alpha\) is the identity on \(A\), so that \(\alpha\) induces isomorphisms of the homotopy sequence of the pair \((A_{\infty}, A)\) onto that of the pair \((\Omega, A)\) and of the singular homology sequence of the pair \((A_{\infty}, A)\) onto that of the pair \((\Omega, A)\). The map \(\alpha\) is defined by means of a distance function, \(\rho\), at \(a^0\), that is a map of \(A\) into the non-negative reals which is zero only at \(a^0\). Such a \(\rho\) may always be defined and the choice of \(\rho\) does not affect the homotopy class of \(\alpha\) and therefore the induced isomorphisms. We now give the construction of \(A_{\infty}\). Let \(A_m\) be the set of sequences of at most \(m\) points of \(A -a^0\); the empty sequence is identified with \(a^0\). Let \(A^m\) be the topological product of \(m\) copies of \(A\). A point of \(A^m\) may be regarded as an infinite sequence \((a_r), a_r \in A\), with \(a_r=a^0\), \(r>m\); then a map \(p_m: A^m \rightarrow A_m\) is defined by associating with each sequence \((a_r)\) the finite subsequence obtained by omitting the terms \(a^0\), and \(A_m\) is given the identification topology induced by \(p_m\). Let \(A_{\infty} = \bigcup_{m \geq 0}^{\cup} A_m\) be given the weak (fine) topology: \(F \subseteq A_{\infty}\) is closed if and only if \(F \cap A_m\) is closed for all \(m\). If \((e_i)\) is the set of cells of \(A\), then the cells of \(A_m\) are \(a^0\) together with product cells \(e_1\times e_2 \times \cdots \times e_r\), \(1 \leq r \leq m \). Thus if \(A=S^n\), an important example, \[ A_m = a^0 \cup e^n \cup e^{2n} \cup \cdots \cup e^{mn}= S^{n} \cup e^{2 n} \cup \cdots \cup e^{m n}, \] and \[ A_{\infty} = S^n \cup e^{2n} \cup \cdots \cup e^{mn} \cup \cdots, \] and \(A_{\infty}\) has the singular homotopy type of \(\Omega\left(S^{n+1}\right)\). The homology of \(A_{\infty}\) is given as follows. Let the sequence of graded groups \(G^1, G^2, \ldots, G^m, \ldots \), be given by \[ G^1 = \sum_{r=1}^{\infty} H_{r}(A), \quad G^m = G^1 \otimes G^{m-1} + G^{1} * G^{m-1}. \] Then \[ \sum_{r=1}^{\infty} H_{r}\left(A_{m}\right) \cong \sum_{n=1}^{m} G^{n}, \quad m=1,2, \ldots, \infty. \] Two further useful features of the reduced product complex \(A_{\infty}\) are mentioned by the author; first the filtration of \(A_{\infty}\) by the subcomplexes \( A_{m} \), and, second, the presence of certain canonical extension properties. Clearly a map \(h: A, a^0 \rightarrow B, b^0\) induces maps \(h^m: A^m \rightarrow B^m\), \(h_m: A_m \rightarrow B_m\), \(h_{\infty}: A_{\infty} \rightarrow B_{\infty}\). More generally, a map \(h: A_m, A_{m-1} \rightarrow B_n, b^0\) admits a combinatorial extension \(h': A_{\infty} \rightarrow B_{\infty}\) which is natural in an obvious sense. The results of this paper are to be applied in a forthcoming paper [On the suspension triad. Ann. Math. (2) 63, 191--247 (1956; Zbl 0071.17002)], by the author. \textit{H. Toda} [Proc. Japan. Acad. 29, 299--304 (1953; Zbl 0053.30201)] has also described a method of replacing the space of loops on a suspension space by a more tractable space within its singular homotopy type.