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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.

Weil homomorphism in non-commutative differential spaces

Jagodziński T., Sasin W.

Demonstratio Mathematica

2005

Vol. 38, nr 2

507--516

In this paper we construct Weil homomorphism in locally free module over a non-commutative differential space, which is a generalization of Sikorski differential space [6]. We consider real case, but the complex case can be done analogusly.

Sikorski and Frolicher CW-complexes compared

Ntumba P. P.

Demonstratio Mathematica

2005

Vol. 38, nr 1

207--221

In this paper we analyze the underlying topological space of a Sikorski CW-complex and the close relationship between Sikorski CW-complexes and Frolicher CW-complexes. Sikorski and Frolicher CW-complexes are analogues of CW-complexes in the categories of differential spaces (a la Sikorski) and Frolicher spaces relatively.