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A generalized homology theory obeys the homotopy, exactness, suspension and wedge axioms, without imposing the ordinary dimension axiom. A represented homology theory is obtained by taking stable homotopy groups after a smash product of spectra.

A topological spectrum $E$ represents the reduced generalized homology theory ${\tilde{E}}_k(X)={\pi }_k(E\wedge {\Sigma }^{\infty }X)$ on based spaces, where the smash product of spectra is derived. It is covariant in $X$.