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A note on linear mappings between function spaces

Jan Baars — 1993

Commentationes Mathematicae Universitatis Carolinae

Arhangel’skiǐ proved that if X and Y are completely regular spaces such that C p ( X ) and C p ( Y ) are linearly homeomorphic, then X is pseudocompact if and only if Y is pseudocompact. In addition he proved the same result for compactness, σ -compactness and realcompactness. In this paper we prove that if φ : C p ( X ) C p ( X ) is a continuous linear surjection, then Y is pseudocompact provided X is and if φ is a continuous linear injection, then X is pseudocompact provided Y is. We also give examples that both statements do not hold...

Equivalence of certain free topological groups

Jan Baars — 1992

Commentationes Mathematicae Universitatis Carolinae

In this paper we give a complete isomorphical classification of free topological groups F M ( X ) of locally compact zero-dimensional separable metric spaces X . From this classification we obtain for locally compact zero-dimensional separable metric spaces X and Y that the free topological groups F M ( X ) and F M ( Y ) are isomorphic if and only if C p ( X ) and C p ( Y ) are linearly homeomorphic.

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