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• W1543138347 abstract "Let U be an open subset of R I = dir lim R', where R denotes the reals. We show that U x R I is homeomorphic to U. Combined with previous work of the author we obtain the corollary that two open subsets of R I are homeomorphic if and only if they have the same homotopy type. Let R denote the reals, and let R = dir lim R'. Here we prove the following. THEOREM 1. If U is an open subset of R ? then U x R ? is homeomorphic to U. In [2, Theorem 11-8] it is shown that if two paracompact, connected R ?-manifolds N, M have the same homotopy type then N x R and M X R ? are homeomorphic. Since open subsets of R ? are paracompact [2, Proposition 111-1] we obtain the following. COROLLARY 2. Two open subsets of R ? are homeomorphic iff they have the same homotopy type. PROOF OF THEOREM 1. We regard R ? as U tRnIn = 1, 2, . .. }, and we identify Rn with Rn x {0} c Rn+k, k > 1, so that Rn c Rn+k c R ??. Let U be an open subset of R ?. We may assume that U is connected. By [2, Proposition 111-2] (or by elementary reasoning) U = U { C, In = 1, 2, . . . } where Cn is compact, Cn c Cn + 1, n > 1, and where a subset V of U is open in U iff V n Cn is open in Cn, n > 1. An elementary argument shows we may assume additionally that Cn c Rn. In what follows the word manifold will be used only for a compact, p.l. (piecewise linear) manifold, possibly with boundary. We observe that if K is any compact set and if K c W where W is open in R n, then there is an n-manifold M such that K c M c W. This manifold may be obtained, for example, by first finding a polyhedron in W containing K and then taking a regular neighborhood of this polyhedron in W. For n > 1 and E > 0 let DF(n, ) = lt UX =(X U Xn)-RnIXil 0). For n 1 Ilet un= U n Rn. Received by the editors October 27, 1975 and, in revised form, February 23, 1976. AMS (MOS) subject classifications (1970). Primary 57A20, 54B99." @default.
• W1543138347 created "2016-06-24" @default.
• W1543138347 creator A5064751800 @default.
• W1543138347 date "1976-02-01" @default.
• W1543138347 modified "2023-10-18" @default.
• W1543138347 title "Open subsets of $Rsp{infty }$ are stable" @default.
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• W1543138347 doi "https://doi.org/10.1090/s0002-9939-1976-0425974-0" @default.
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