1

I have read in several books that a connected paracompact Hausdorff one-dimensional manifold without boundary points is homeomorphic to the circle if it is compact. However I dont figure out how to prove it.

Rougly speaking, I if the one-dimension manifold is connected, it is homeomorphic to a interval in the real line, but an interval is not homeomorphic to the circle, Am I wrong?

Mani
  • 83
  • 7
  • 1
    Look at Guillemin and Pollack's Differential Topology or Milnor's book for a proof if you consider smooth maifolds. For topological manifolds i think i've seen in Lee's Introduction to Topological Manifolds. – Kelvin Lois May 10 '20 at 14:36
  • if the manifold is paracompact but not compact it is homeomorphic to $\Bbb R$, if it is however compact, to $S^1$. So just two options in one-dimension (for connected manifolds, of course). – Henno Brandsma May 10 '20 at 15:20
  • A $1$-dimensional manifold is locally homeomorphic to an open interval in $\Bbb R$. But you forgot about the hypothesis of compactness. – Ted Shifrin May 10 '20 at 19:12
  • A space is compact provided each its open cover has a finite subcover. The circle is compact, the real line is not. – Alex Ravsky May 28 '20 at 01:41

0 Answers0