Continuous selections and 𝐶-spaces

Abstract

A characterization of paracompact <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -spaces via continuous selections avoiding <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">Z_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -sets is given. The result is applied to prove a countable sum theorem for paracompact <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -spaces, and to obtain a new partial solution of a question raised by E. Michael.