some of the so called smallness conditions in algebra as well as in category theory, are important and interesting for their own and also tightly related to injectivity, are essential boundedness, cogenerating set, and residual smallness.in this overview paper, we first try to refresh these smallness condition by giving the detailed proofs of the results mainly by bernhard banaschewski and walt...

Let h≥2 and let A=(A1,…,Ah) be an h-tuple of sets integers. For nonzero integers c1,…,ch, consider the linear form φ=c1x1+c2x2+⋯+chxh. The representation function RA,φ(n) counts number h-tuples (a1,…,ah)∈A1×⋯×Ah such that φ(a1,…,ah)=n. A is a φ-Sidon system multiplicity g if RA,φ(n)≤g for all n∈Z. every positive integer g, Fφ,g(n) denote largest q there exists withAi⊆[1,n]and|Ai|=q i=1,…,h. It ...

We determine the cut locus of arbitrary non-simply connected, compact and irreducible Riemannian symmetric space explicitly, and compute injectivity radius and diameter for every type of them.