Hurwitz' theorem implies Rouché's theorem

AbstractIt is well known that Hurwitz's theorem is easily proved from Rouché's theorem. We show that conversely, Rouché's theorem is readily proved from Hurwitz' theorem. Since Hurwitz' theorem is easily proved from the formula giving the number of roots of an analytic function, our result thus gives also a simple proof of Rouché's theorem

Nest and Complete Accumulation Point Compactness ot the Product of Topological Spaces

  Based on the definition of nest compactness (Ie., the intersection of a nest of nonempty closed sets is nonempty) we show that the product of the two nest compáct topological spaces is nest compact, and, this without invoking the compactness of the product of two compact topological spaces based on the classical definition of compactness (i.e., every open cover has a finit subcover). The same is done based on the definition of complete accumulation point compactness. The latter, by Remark 5, extends easily to the infinite products of topological spaces. &nbsp

Nest and Complete Accumulation Point Compactness ot the Product of Topological Spaces

  Based on the definition of nest compactness (Ie., the intersection of a nest of nonempty closed sets is nonempty) we show that the product of the two nest compáct topological spaces is nest compact, and, this without invoking the compactness of the product of two compact topological spaces based on the classical definition of compactness (i.e., every open cover has a finit subcover). The same is done based on the definition of complete accumulation point compactness. The latter, by Remark 5, extends easily to the infinite products of topological spaces. &nbsp

Rings of real-valued functions and the finite subcovering property

Let C be a ring of (not necessarily bounded) real-valued functions with a common domain X such that C includes all the constant functions and if f   and lt; C then | f | ε C