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Suppose we have a CW-complex $X$ with a 0-cell $e^0$. Is the union of all the cells (of higher dimensions) for which $e^0$ is a boundary point open in $X$?

I don't know if it has a name, but a similar thing for vertices of a polyhedron in the Russian literature is called "star".

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    $\begingroup$ No. In a CW-complex you can attach cells in a weird way. For example, take S^1 (with one zero cell and one one-cell) and attach a two cell by sending S^1 to one interior point of the one-cell). Then the star of the zero cell would be S^1, which is not open. $\endgroup$
    – HenrikRüping
    Commented Mar 13 at 14:10
  • $\begingroup$ @HenrikRüping Thank you for the example but will this be still false for the star with punctured zero cell? $\endgroup$
    – brattok
    Commented Mar 13 at 15:11

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