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Proceedings of the American Mathematical Society

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A locally simply connected space and fundamental groups of one point unions of cones


Author: Katsuya Eda
Journal: Proc. Amer. Math. Soc. 116 (1992), 239-249
MSC: Primary 55Q20; Secondary 57M05
DOI: https://doi.org/10.1090/S0002-9939-1992-1132409-0
MathSciNet review: 1132409
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Abstract: Let CX be the cone over a space $ X$. Let a space $ X$ be first countable at $ x$, then the following are equivalent: (1) $ X$ is locally simply connected at $ x$; (2) $ {\pi _1}\left( {\left( {X,x} \right) \vee \left( {X,x} \right),x} \right)$ is naturally isomorphic to the free product $ {\pi _1}\left( {X,x} \right) * {\pi _1}\left( {X,x} \right)$; (3) $ {\pi _1}\left( {\left( {CX,x} \right) \vee \left( {CX,x} \right),x} \right)$ is trivial. There exists a simply connected, locally simply connected Tychonoff space $ X$ with $ x \in X$, such that $ \left( {X,x} \right) \vee \left( {X,x} \right)$ is not simply connected.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1132409-0
Keywords: Fundamental group, locally simple, first countable, one point union, cone
Article copyright: © Copyright 1992 American Mathematical Society