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I've been wondering about "topologically equivalent" for some time now. For example:

$S^1$ is "topologically equivalent" to $\mathbb{R}P^1$.

I see that they are homotopy equivalent. But are they also homeomorphic? Probably yes.

Is there a failsafe way to determine whether in a given case "topologically equivalent" means "homeomorphic" or "homotopy equivalent"? Thanks for your help!

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    Or, using identifications, consider the upper-half of $S^1$, after the identifications have been made, you still need to identify the tips. You can do this with the map $e^it$ defined on the unit interval, which sends the points in the interior to interior point, and collapses 0 with 1. You then get the collection of points (cos2Pit,sin2Pit), t=0 to 1.2011-06-20

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It's ambiguous, and that's why you shouldn't use it. I've only seen the term used in popular texts to explain topology without explaining topology, and in my experience it could mean at least three things the way it's used:

  • homeomorphic
  • homotopy equivalent
  • isotopic.

In this case, $S^1$ and $\mathbb{R}P^1$ are homeomorphic. The explicit homeomorphism is not difficult to construct (it comes from the $2$-to-$1$ cover $S^1 \to S^1$).

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    Hatcher does it : (2011-06-24