We note that the definition of R -functions depends on the choice of a certain surjection and pose the problem of the construction of a function of two variables that is not an R -function for any choice of a surjective mapping. It is shown that the function x 1 x 2 1 possesses ...
Proof(a)ppis surjective is obvious. (b) Proveppis continuous. ppis a piecewise function comprised of two partsp1=xp1=xwithx∈[0,1])x∈[0,1])andp2=x−1p2=x−1withx∈[2,3]x∈[2,3]. We extend the domains and ranges ofp1p1andp2p2toRRand obtain two continuous functions~p1p~an...
It is also clear from the diagram that Ho(M) → Vect is surjective on hom-sets. We will prove below that it is actually an equivalence. 2.3. Homotopies. In model categories it is more common to deal with homotopies in terms of cylinder objects rather than path objects, as the former ...
The example is similar to an example given by Nachbin, where he shows that not every metrizable space is a (C)-space, and it answers a question of Dineen. It is also an example of a locally convex space, which is not a surjective limit of metrizable spaces or a σ-convex space....