algebraic variety

If 𝒲 is a locally closed subset (i.e., the intersection of open and closed sets) of an algebraic variety, it becomes an algebraic variety in a natural manner since the germs of regular functions at x∈𝒲 axe taken to be the germs of functions on 𝒲 induced by functions in the stalk Rₓ. The definitions of irreducibility and local rings of subvarieties for algebraic varieties are given in the same manner as before. From now on, by a variety is meant an algebraic variety. Any variety (X,R), by definition, is a local-ringed space.