Factorizing \(\tau \)-spectra of mappings (Q602034)

In this paper, the term spectrum of spaces means an inverse spectrum, or what is the same, an inverse system of spaces. A spectrum of mappings refers to a morphism of a spectrum of spaces. Using the notion, mapping of mappings, the idea of a spectrum of mappings, similar to that of a spectrum of spaces is given. For spectra of mappings, the author defines the concept of a \(\tau\)-spectrum of mappings factorizing in a special sense. A version of the Spectral Theorem (Proposition 1.3) is proved for such spectra. To a given indexed collection \textbf{F} of mappings is associated a factorizing \(\tau\)-spectrum whose mappings are Containing Mappings for \textbf{F} as constructed in [\textit{S. Iliadis}, Universal spaces and mappings. North-Holland Mathematics Studies 198. Amsterdam: Elsevier. (2005; Zbl 1072.54001)]. These associated \(\tau\)-spectra and the corresponding version of the Spectral Theorem imply that for a given \textbf{F} any so-called ``natural'' factorizing \(\tau\)-spectrum for \textbf{F} contains a cofinal and \(\tau\)-closed subspectrum whose mappings are Containing Mappings for \textbf{F}.