normal doesn't imply paracompact












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I'm looking for some examples which could show that normal topological space doesn't imply the space is paracompact.



Thanks in advance.










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    0












    $begingroup$


    I'm looking for some examples which could show that normal topological space doesn't imply the space is paracompact.



    Thanks in advance.










    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      I'm looking for some examples which could show that normal topological space doesn't imply the space is paracompact.



      Thanks in advance.










      share|cite|improve this question











      $endgroup$




      I'm looking for some examples which could show that normal topological space doesn't imply the space is paracompact.



      Thanks in advance.







      general-topology examples-counterexamples separation-axioms paracompactness






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      edited Dec 9 '18 at 3:23









      Eric Wofsey

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      asked Jan 22 '18 at 17:32









      Chun GanChun Gan

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          $begingroup$

          The space $omega_1$, the set of countable ordinals, in the order topology, is hereditarily normal (as all ordered spaces), countably (para)compact, collectionwise normal, but not paracompact, because a paracompact countably compact is compact, which $omega_1$ is not.



          The nice resource $pi$-base has some more examples, some derived from this one (like the long line), others like Dowker spaces (a normal not countably paracompact space, which are harder to come by).



          Bing's spaces like "G" (nicely explained here and "H" (see here) are also classics, introduced to explore the gap between normality-like properties and paracompact-like properties. I think $omega_1$ is the easiest example though.






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            An example is built here, have you read it:
            https://projecteuclid.org/download/pdf_1/euclid.bams/1183549052






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              2 Answers
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              2 Answers
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              $begingroup$

              The space $omega_1$, the set of countable ordinals, in the order topology, is hereditarily normal (as all ordered spaces), countably (para)compact, collectionwise normal, but not paracompact, because a paracompact countably compact is compact, which $omega_1$ is not.



              The nice resource $pi$-base has some more examples, some derived from this one (like the long line), others like Dowker spaces (a normal not countably paracompact space, which are harder to come by).



              Bing's spaces like "G" (nicely explained here and "H" (see here) are also classics, introduced to explore the gap between normality-like properties and paracompact-like properties. I think $omega_1$ is the easiest example though.






              share|cite|improve this answer











              $endgroup$


















                2












                $begingroup$

                The space $omega_1$, the set of countable ordinals, in the order topology, is hereditarily normal (as all ordered spaces), countably (para)compact, collectionwise normal, but not paracompact, because a paracompact countably compact is compact, which $omega_1$ is not.



                The nice resource $pi$-base has some more examples, some derived from this one (like the long line), others like Dowker spaces (a normal not countably paracompact space, which are harder to come by).



                Bing's spaces like "G" (nicely explained here and "H" (see here) are also classics, introduced to explore the gap between normality-like properties and paracompact-like properties. I think $omega_1$ is the easiest example though.






                share|cite|improve this answer











                $endgroup$
















                  2












                  2








                  2





                  $begingroup$

                  The space $omega_1$, the set of countable ordinals, in the order topology, is hereditarily normal (as all ordered spaces), countably (para)compact, collectionwise normal, but not paracompact, because a paracompact countably compact is compact, which $omega_1$ is not.



                  The nice resource $pi$-base has some more examples, some derived from this one (like the long line), others like Dowker spaces (a normal not countably paracompact space, which are harder to come by).



                  Bing's spaces like "G" (nicely explained here and "H" (see here) are also classics, introduced to explore the gap between normality-like properties and paracompact-like properties. I think $omega_1$ is the easiest example though.






                  share|cite|improve this answer











                  $endgroup$



                  The space $omega_1$, the set of countable ordinals, in the order topology, is hereditarily normal (as all ordered spaces), countably (para)compact, collectionwise normal, but not paracompact, because a paracompact countably compact is compact, which $omega_1$ is not.



                  The nice resource $pi$-base has some more examples, some derived from this one (like the long line), others like Dowker spaces (a normal not countably paracompact space, which are harder to come by).



                  Bing's spaces like "G" (nicely explained here and "H" (see here) are also classics, introduced to explore the gap between normality-like properties and paracompact-like properties. I think $omega_1$ is the easiest example though.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Jan 22 '18 at 22:34

























                  answered Jan 22 '18 at 20:13









                  Henno BrandsmaHenno Brandsma

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                  107k347114























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                      $begingroup$

                      An example is built here, have you read it:
                      https://projecteuclid.org/download/pdf_1/euclid.bams/1183549052






                      share|cite|improve this answer









                      $endgroup$


















                        1












                        $begingroup$

                        An example is built here, have you read it:
                        https://projecteuclid.org/download/pdf_1/euclid.bams/1183549052






                        share|cite|improve this answer









                        $endgroup$
















                          1












                          1








                          1





                          $begingroup$

                          An example is built here, have you read it:
                          https://projecteuclid.org/download/pdf_1/euclid.bams/1183549052






                          share|cite|improve this answer









                          $endgroup$



                          An example is built here, have you read it:
                          https://projecteuclid.org/download/pdf_1/euclid.bams/1183549052







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Jan 22 '18 at 18:06









                          Arnaud MortierArnaud Mortier

                          19.8k22260




                          19.8k22260






























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