Type: Article
Publication Date: 1984-01-01
Citations: 2
DOI: https://doi.org/10.1090/s0002-9939-1984-0735577-0
Compact almost continuous retracts of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Superscript n Baseline left-parenthesis n greater-than-or-slanted-equals 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>β©Ύ<!-- β©Ύ --></mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{I^n}(n \geqslant 2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> do not separate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{E^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Some other results that hold for continuous functions are also shown to hold for almost continuous functions. A result in [<bold>5</bold>] giving sufficient conditions for a set to be an almost continuous retract of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{I^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is examined further, and a method of constructing some almost continuous retracts of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{I^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is given.