I'm using the following version of Half-space Theorem:
$\textbf{Theorem}$(Half-space) A connected, immersed proper, nonplanar minimal surface $M$ in $\mathbb{R}^3$is not contained in a halfspace.
I supposedly proved this theorem using the closed and complete hypothesis instead of immersed proper.
Is the "immersed proper" hypothesis necessary?
The original statement is found in the paper "The strong halfspace theorem for minimal surfaces" by D. Hoffman and W. H. Meeks, III, 1990. In the statement they use the hypothesis of immersed proper and allow the surface to be "possibly branched".
Does the fact that the surface is "possibly branched" need the proper immersed hypothesis?
Can someone help me understand this better?
Follow the link in the paper below http://www.math.jhu.edu/~js/Math748/hoffman-meeks.halfspace.pdf