A topological characterization of countable functionals
Ullman, Arthur William James
Robinson, John Alan
Master of Arts
This topology is called the neighborhood topology. It then follows that a functional F is countable if it is continuous on the neighborhood topology. This then is our topological characterization. The topology is shown to be metrizable by observing that the base is both countable and closed. The above definitions and results are extended to include all functionals of pure type. There is a uniform method for transforming any functional into one of pure type but it is also shown that there is no uniform method of topologizing impure spaces in such a way that the countable functionals are characterized as precisely those which are continuous.