A nonstandard approach to realcompactness with applications to the equation ν(X × Y) = ν(X) × ν(Y)

The set psns∗X of all pseudonearstandard points and its relationship to topological concepts will be systemically investigated. For example we obtain nonstandard characterizations for the property that the realcompactification ν(X) is locally compact or a bi-k-space. The study of the equation psns∗(X × Y) = psns∗X × psns∗Y is equivalent to the question whether the extension of the identity map id: X × Y → ν(X) × ν(Y) to the space ν(X × y) is a biquotient map and sufficient conditions are proved. Moreover we show that this question is not equivalent to the validity of the equation ν(X × Y) = ν(X) × ν(Y) under the assumption of the existence of a measurable cardinal. If ν(X) and ν(Y) are bi-k-spaces both equations are equivalent to the property that the product of two bounded subsets is bounded.