I read that every open subset $A$ of a manifold $M$ is a submanifold (it is a manifold with the induced topology by $M$). If I understand correctly, the argument is that, for an element $x \in A$, one can considerer a chart $(U,f)$ in $M$, and clearly its cutting with $A$ gives a chart in $A$. My question is the following: why every subset $B$ of a manifold isn't a submanifold? The cutting of a chart of $M$ with $B$ gives a chart on $B$ with the induced topology, because the open sets in $B$ are the open sets in $M$ cut with $B$. I realize that something is wrong in my reasoning because it is know that not every subset of a submanifold is a manifold, but I do not know what exactly. Can someone help me?
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The charts need to be homeomorphisms with patches of open subsets of $B$. The open subsets of an arbitrary subset aren't necessarily locally Euclidean. – Adam Hughes Nov 20 '14 at 20:53
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You need a transversality condition. This is why submersions are important. – user40276 Nov 20 '14 at 22:05
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A chart must be homeomorphic to Euclidean space, so intersecting a chart with a subspace need not give you a chart in the subspace, even if you cut it down. Consider the example of the rational numbers as a subset of the real numbers.
Matt Samuel
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