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So I have been doing some topology and I came up with this question, and can't find a simple way to prove it. I can kinda make the argument that if we have two points in $M-N$, the are connected by a path in $M^C$, and you should be able to project this onto $M$ and let it into the neighborhoods $M-N$. I am able to prove this is locally path connected, but keep getting stuck trying the to prove it rigorously. I am mainly interested in the case that we are in $\mathbb{R}^n$. Could anyone help me out? If it is not true, is it true if $M$ is a manifold?

Any advice is helpful, thanks!

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