説明
<jats:p>This paper continues the study of continuous selections begun in (13; 14; 15) and the expository paper (12). The purpose of these papers, which is described in detail in the introduction to (13), can be summarized here as follows. If <jats:italic>X</jats:italic> and <jats:italic>Y</jats:italic> are topological spaces, and <jats:italic>ϕ</jats:italic> a function (called a <jats:italic>carrier)</jats:italic> from <jats:italic>X</jats:italic> to the space <jats:italic>2<jats:sup>Y</jats:sup></jats:italic> of non-empty subsets of F, then a <jats:italic>selection</jats:italic> for ϕ is a continuous f: <jats:italic>X → Y</jats:italic> such that f(x) ∈ <jats:italic>ϕ(x)</jats:italic> for every <jats:italic>x</jats:italic> ∈ <jats:italic>X</jats:italic>. For reasons which are explained in (13), we restrict our attention to carriers which are <jats:italic>lower semi-continuous</jats:italic> (l.s.c), in the sense that, whenever <jats:italic>U</jats:italic> is open in Y, then <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0008414X00003424_inline1" /> is open in <jats:italic>X</jats:italic>. Our purpose in these papers is to find conditions for the existence and extendability of selections.</jats:p><jats:p>The principal purpose of <jats:italic>this</jats:italic> paper is to generalize the following result, which is half of the principal theorem (Theorem 3.2˝) of (13) (and is repeated as Theorem I of (12)).</jats:p>
収録刊行物
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- Canadian Journal of Mathematics
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Canadian Journal of Mathematics 11 556-575, 1959
Canadian Mathematical Society
