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Remarks on smooth real-compactness for Sikorski spaces

Cukrowski M. J., Stronkowski M. M.

Demonstratio Mathematica

2014

Vol. 47, nr 2

465--473

It is known that every Sikorski space with the countably generated differential structure is smoothly real-compact. It means that every homomorphism from its differential structure, which forms a ring of smooth real-valued functions into the ring of real numbers, is an evaluation. This result is sharp: there is a non-smoothly real-compact Sikorski space with the differential structure which is not countably generated. We provide an easy example demonstrating this. By modifying this example we are able to show a certain shortcoming of the generator embedding, comparing to the canonical embedding, for Sikorski spaces. Finally, we note that a homomorphism from the ring of smooth functions of a Sikorski space into the ring of real numbers is an evaluation if and only if it is continuous.

On real-valued homomorphisms in countably generated differential structures

Cukrowski M. J., Pasternak-Winiarski Z., Sasin W.

Demonstratio Mathematica

2012

Vol. 45, nr 3

665-676

Real valued homomorphisms on the algebra of smooth functions on a differential space are described. The concept of generators of this algebra is emphasized in this description.