An H-space is a topological space X with basepoint e and a multiplication map m: X 2 = XXX->X such that e is a homotopy identity element, (We take all maps and homotopies in the based sense. We use k-topologies throughout in order to avoid spurious topological difficulties. This gives function spaces a canonical topology.) We call X a monoid if m is associative and e is a strict identity. In the literature there are many kinds of ü-space: homotopyassociative, homotopy-commutative, ^««-spaces

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... , etc. In the last case part of the structure consists of higher coherence homotopies. In this note we introduce the concept of homotopy-everything H-space {E-space for short), in which all possible coherence conditions hold. We can also define £-maps (see §4). Our two main theorems are Theorem A, which classifies E-spaces, and Theorem C, which provides familiar examples such as BPL. Many of the results are folk theorems. Full details will appear elsewhere. A space X is called an infinite loop space if there is a sequence of spaces X n and homotopy equivalences X n c^tiX n+ i for n^O, such that X = Xo. THEOREM A. A CW-complex X admits an E-space structure with TQ(X) a group if and only if it is an infinite loop space. Every E-space X has a (i 'classifying space" BX, which is again an E-space.