We define a function f : X → Y to be slightly β-continuous if for every clopen set V of Y , f−1(V)⊂ Cl(Int(Cl(f−1(V)))). We obtain several properties of such a function. Especially, we define the notion of ultra-regularizations of a topology and obtain interesting characterizations of slightly β-continuous functions by using it. 2000 Mathematics Subject Classification. 54C08.

We introduce a new class of multifunctions called a slightly m-continuous multifunction which is a generalization of both slightly continuous functions [10] and m-continuous multifunctions [33]. In this paper we obtain several properties of such a multifunction.

The classic renaming protocol of Moir and Anderson [4] uses a network of Θ(n) splitters to assign unique names to n processes with unbounded initial names. We show how to reduce this bound to Θ(n) splitters.