Relationships between continuum neighborhoods in inverse limit spaces and separations in inverse limit sequences

Author:

Davis Harvey S.

Abstract

The main result of this paper is the following theorem. Let { X α , f α β , α , β I } \{ {X_\alpha },{f_{\alpha \beta }},\alpha ,\beta \in I\} be an inverse system of compact Hausdorff spaces and continuous onto maps with inverse limit X. Let p X p \in X and A be closed in X. There exists a continuum neighborhood of p disjoint from A if and only if there exists α I \alpha \in I and disjoint sets U and V open in X α {X_\alpha } , neighborhoods respectively of p α {p_\alpha } and A α {A_\alpha } such that for all β α , f α β 1 ( U ) \beta \geqslant \alpha ,f_{\alpha \beta }^{ - 1}(U) lies in a single component of X β f α β 1 ( V ) {X_\beta } - f_{\alpha \beta }^{ - 1}(V) . This is Theorem B of the text.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference5 articles.

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2. A characterization of locally connectedness by means of the set function 𝑇;Bennett, Donald E.;Fund. Math.,1974

3. N. Bourbaki, General topology. Parts 1 and 2, Addison-Wesley, Reading, Mass., 1966. MR 34 # 5044a, 5044b.

4. Properties of the set functions 𝑇ⁿ;Davis, H. S.;Portugal. Math.,1962

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