2
$\begingroup$
  1. Does the dimension of a manifold depend on the topology? That is, can I endow a set with a topology $T$ and get an $n$-dimensional manifold, and endow the exact same set with another toplogy $T'$ and get an $m$-dimensional manifold, with $n\neq m$?

  2. Can a curve like this:

$\hskip1.3in$enter image description here

be given the topology of a 1-manifold? Or a curve like this cannot be a manifold with any topology?

  • 3
    It seems hopeless to characterize what you can do with a bare _set_. $\mathbf R$ and $\mathbf R^2$ have the same cardinality, for example.2012-04-15
  • 0
    OK so you are saying that those questions are impossible to prove or disprove?2012-04-15
  • 2
    @kein Exapnding on Dylan comment: For each $n$ you can give a topology to $\mathbb R$ so that $\mathbb R$ is an $n$-manifold.2012-04-15
  • 0
    Yep, that is correct. If S is a bijection between R and R^n, you can give R the topology induced by S (the open sets of R are the images of S which are open on R^n in the usual sense), and with that S is an homeomorphism. Thanks for helping me clarify this topic.2012-04-15

1 Answers 1