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Let $X$ be a topological $n$-manifold and $N$ a $d$-submanifold of $X$, ($d\leq n$), then under what conditions on $X$ and $N$ do we have that the complement $X-N$ is again a manifold and what is the dimension of the submanifold $X-N$?

For example if $N$ is a submanifold that is closed as a subset of $X$ then $X-N$ is an open subset of $X$ hence it is a submanifold of $X$ with the same dimension $n$. but what about the case of $N$ not being a closed subset of $X$?

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What precisely is your definition of a manifold? (Do you allow boundary or corners?) I ask because the answer to your question depends a bit on what you consider as a manifold. –  Willie Wong Oct 12 '12 at 11:35
    
that's my question, what $X$ might be for this to happen. –  palio Oct 12 '12 at 11:38
    
What if $X$ is a closed manifold? that is it is compact without boundary? –  palio Oct 16 '12 at 7:28
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