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@Mariano: In my situation the function is not a Morse function and the curves of level are connected by paths, I think that this hypothesis about U are enough, but I can't proof this affirmation.

@Anton: I don't know if the Whitehead manifold have has a function with this properties.

I want to proof that the compact sets $K_m = U-f^{-1}([0,m])$ (assuming that $0$ is the minimum value of $f$) are simply-connected, this will implies that $U$ is simply-connected to infinite.

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@Mariano: In my situation the function is not a Morse function and the curves of level are connected by paths, I think that this hypothesis about U are enough, but I can't proof this affirmation.

@Anton: I don't know if the Whitehead manifold have a function with this properties.

I want to proof that the compact sets $K_m = U-f^{-1}([0,m])$ (assuming that $0$ is the minimum value of $f$) are simply-connected, this will implies that $U$ is simply-connected to infinite.