  In this paper we will extend the product of spectral triples to a product of
semifinite spectral triples. We will prove that finite summability and
regularity are preserved under taking products. Connes and Marcolli constructed
for each $z\in(0,\infty)$ a type ${\rm II}_\infty$-semifinite spectral triple
which can be considered as a geometric space of dimension $z$. A small adaption
of their construction yields a type ${\rm I}$-semifinite spectral triple. We
will investigate the properties of these semifinite spectral triples. At the
same time we will also avoid the need for an infra-red cutoff to compute the
dimension spectrum. Using this collection of semifinite spectral triples and
the product of semifinite spectral triples one can construct a mathematical
tool for dimensional and zeta-function regularisation in quantum field theory.
