Suppose I have a $1$-dimensional manifold $M$ and within it an open set $U$ such that $U$ is homeomorphic to $S^1$. Can I deduce from that that $U$ is also a closed set inside $M$?
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As $S^1$ is compact, $U$ is compact too (as it is homeomorphic) and assuming the manifold is Hausdorff (as is usual) then indeed $U$ is indeed closed. And as noted in the comments, if $M$ is connected this would imply $M = U$.