A regular inductive limit of sequentially
complete spaces is sequentially complete. For the converse of this theorem
we have a weaker result: if ind <mml:math alttext="$E_n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> is sequentially complete inductive limit,
and each constituent space <mml:math alttext="$E_n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> is closed in ind <mml:math alttext="$E_n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math>, then ind 
<mml:math alttext="$E_n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>E</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> is <mml:math alttext="$\alpha$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#945;</mml:mi></mml:math>-regular.

International Journal of Mathematics and Mathematical Sciences

Sequential completeness of inductive limits

Abstract

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