Closure of an open cell in a CW complex












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Given a CW complex $X$ and any of its cells $e$, the closure $bar{e}$ in $X$ is covered by finitely many open cells by "C". Can we prove that $bar{e}$ is exactly a union of open cells, or, $(bar{e}-e)$ a union of open cells of strictly lower dimensions? If not, is there any counterexample?






general-topology algebraic-topology







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    1












    $begingroup$


    Given a CW complex $X$ and any of its cells $e$, the closure $bar{e}$ in $X$ is covered by finitely many open cells by "C". Can we prove that $bar{e}$ is exactly a union of open cells, or, $(bar{e}-e)$ a union of open cells of strictly lower dimensions? If not, is there any counterexample?






    general-topology algebraic-topology







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      Given a CW complex $X$ and any of its cells $e$, the closure $bar{e}$ in $X$ is covered by finitely many open cells by "C". Can we prove that $bar{e}$ is exactly a union of open cells, or, $(bar{e}-e)$ a union of open cells of strictly lower dimensions? If not, is there any counterexample?






      general-topology algebraic-topology







      share|cite|improve this question









      $endgroup$




      Given a CW complex $X$ and any of its cells $e$, the closure $bar{e}$ in $X$ is covered by finitely many open cells by "C". Can we prove that $bar{e}$ is exactly a union of open cells, or, $(bar{e}-e)$ a union of open cells of strictly lower dimensions? If not, is there any counterexample?






      general-topology algebraic-topology




      general-topology algebraic-topology






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 26 '18 at 8:25









      Smart YaoSmart Yao

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      1369






















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          $begingroup$

          In general $overline e$ won't be a union of open cells.



          Simple example. One $0$-cell $e_0$, one $1$-cell $e_1$
          and one $2$-cell $e_2$. Attach $e_1$ to $e_0$ making an $S^1$. Now attach $e_2$
          to this $S_1$ by mapping the boundary of the unit disc to a point $P$ on $S^1$ that
          isn't $e_0$. Then $overline{e_2}=e_2cup{P}$ and that isn't a union of
          open cells.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Does it imply the so-called $delta$-complex if the condition in the question is imposed?
            $endgroup$
            – Smart Yao
            Jan 5 at 1:00











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          1 Answer
          1






          active

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          active

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          active

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          $begingroup$

          In general $overline e$ won't be a union of open cells.



          Simple example. One $0$-cell $e_0$, one $1$-cell $e_1$
          and one $2$-cell $e_2$. Attach $e_1$ to $e_0$ making an $S^1$. Now attach $e_2$
          to this $S_1$ by mapping the boundary of the unit disc to a point $P$ on $S^1$ that
          isn't $e_0$. Then $overline{e_2}=e_2cup{P}$ and that isn't a union of
          open cells.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Does it imply the so-called $delta$-complex if the condition in the question is imposed?
            $endgroup$
            – Smart Yao
            Jan 5 at 1:00
















          2












          $begingroup$

          In general $overline e$ won't be a union of open cells.



          Simple example. One $0$-cell $e_0$, one $1$-cell $e_1$
          and one $2$-cell $e_2$. Attach $e_1$ to $e_0$ making an $S^1$. Now attach $e_2$
          to this $S_1$ by mapping the boundary of the unit disc to a point $P$ on $S^1$ that
          isn't $e_0$. Then $overline{e_2}=e_2cup{P}$ and that isn't a union of
          open cells.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Does it imply the so-called $delta$-complex if the condition in the question is imposed?
            $endgroup$
            – Smart Yao
            Jan 5 at 1:00














          2












          2








          2





          $begingroup$

          In general $overline e$ won't be a union of open cells.



          Simple example. One $0$-cell $e_0$, one $1$-cell $e_1$
          and one $2$-cell $e_2$. Attach $e_1$ to $e_0$ making an $S^1$. Now attach $e_2$
          to this $S_1$ by mapping the boundary of the unit disc to a point $P$ on $S^1$ that
          isn't $e_0$. Then $overline{e_2}=e_2cup{P}$ and that isn't a union of
          open cells.






          share|cite|improve this answer











          $endgroup$



          In general $overline e$ won't be a union of open cells.



          Simple example. One $0$-cell $e_0$, one $1$-cell $e_1$
          and one $2$-cell $e_2$. Attach $e_1$ to $e_0$ making an $S^1$. Now attach $e_2$
          to this $S_1$ by mapping the boundary of the unit disc to a point $P$ on $S^1$ that
          isn't $e_0$. Then $overline{e_2}=e_2cup{P}$ and that isn't a union of
          open cells.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 26 '18 at 8:42

























          answered Dec 26 '18 at 8:30









          Lord Shark the UnknownLord Shark the Unknown

          106k1161133




          106k1161133












          • $begingroup$
            Does it imply the so-called $delta$-complex if the condition in the question is imposed?
            $endgroup$
            – Smart Yao
            Jan 5 at 1:00


















          • $begingroup$
            Does it imply the so-called $delta$-complex if the condition in the question is imposed?
            $endgroup$
            – Smart Yao
            Jan 5 at 1:00
















          $begingroup$
          Does it imply the so-called $delta$-complex if the condition in the question is imposed?
          $endgroup$
          – Smart Yao
          Jan 5 at 1:00




          $begingroup$
          Does it imply the so-called $delta$-complex if the condition in the question is imposed?
          $endgroup$
          – Smart Yao
          Jan 5 at 1:00


















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