A spectral sequence associated with a continuous map

A spectral sequence is defined for a closed map of finite multiplicity which coincides with the Cartan-Grothendieck spectral sequence in the case of a map onto a quotient space by a finite group acting freely $[1,2]$. It is proved that the resolution by means of which the spectral sequence is defined can be described within the framework of the so-called theory of triples. A definition of this sequence is given for an arbitrary continuous map. It is shown that the spectral sequences of coverings are the spectral sequences of special continuous maps.