Kat Etov's Lemma and Monotonically Normal Spaces
نویسنده
چکیده
An interpolation lemma due to Kat etov is simpliied in order to make it more general and easier to use. Several results are established to illustrate applications of this lemma, including a characterization of monotonically normal spaces and a topology-free insertion theorem. We also prove that a topological space has an order-preserving insertion if and only if it is dominated by a collection of closed subspaces, each of which has an order-preserving insertion. The well-known insertion theorem of Kat etov((5],,6]) characterizes normal topological spaces in terms of continuous functions.(See Corollary 2 below.) His proof is based on two interpolation lemmas, and despite their rather complicated hypothesis, they have been used by various authors(e.g. 9], 10], 11]). We state and prove alternative formulations of these lemmas, Theorem 1 and 2 below. Applications and examples are given that include the characterization of normal spaces mentioned above, and we show that the characterization of monotonically normal spaces proved by Kubiak((7]) is a corollary of these versions of Kat etov's lemmas. It is also shown that the \topology-free" insertion theorem of Blair and Swardson ((1]) follows from the Kat etov's lemmas. The last theorem of the paper show that a space has an order-preserving insertion if and only if the space is dominated by a collection of closed subspaces, each of which has an order-preserving insertion. An immediate consequence of this is the result of Miwa((13]) that a space is monotonically normal if and only if it is dominated by a collection of monotonically normal subspaces. Let hR; i denote a partial order on a set R, and whenever innma and suprema are considered, they are taken with respect to. We say that hR; i has least bound property if, for any nonempty countable subset A of R, sup A ex
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