Topological transform of singular points?

[SW VandeCarr]

Are singular points necessarily mapped to singular points under topological transformations? A specific example would a 2-space deformation of a triangle to any closed string with no cross over points. Would the three singular points of the triangle be necessarily mapped to three singular points on the closed string?

 

[SW VandeCarr]

I'm not a professional mathematician or student. Nevertheless I believe I have a legitimate question and practical reasons for knowing the answer. My readings have thrown some light on the question but I would still appreciate an answer:

A triangle is smoothly deformed to a closed loop string without cross over points in 2 space.

Will the three singular points:

1. Be necessarily be mapped to three singular points on the string?

2. If not, can they be mapped to just one or two points on the string?

I am not interested in annihilation techniques other "treatments" of singular points in topological transformations. I stongly suspect #2 is correct, but I would like confirmation.