Cover map of the figure eight

Question

During an exam I was supposed to find a connected (and non trivial) cover map of the figure eight. In particular, each point in the figure eight has to have infinite pre-image. I know that there is a ‘standard’ one in $R^2$ that is not hard at all but during the exam for some reason I forgot about it or I was not able to build it again. Since I had fun thinking about it, I am still curious to know if it works or not.

I came up with an alternative idea. Please let me know how you could formalize this idea in a better way or if there is any flaw in the reasoning :)

1. Start with a triangle (full line) and map each segment to the ‘left circle’. Do you need an ‘orientation’ on the segments when you define this map?
2. For each of the vertices of the triangle in previous step, construct another triangle (dashed line) and map each of its segments to the ‘right circle’.

Then, continue with the same pattern.

I think I missed the orientation but maybe the overall concept work. What do you think? From my reasoning it seems a covering space since it satisfies the required properties.

Answer 1

Yes, that looks like a correct idea.

Of course, whenever one says something like "continue with the same pattern", it raises the question of "What's the pattern?". To give a good answer to such a question, you must give a rigorous description of the pattern, and the best way to do that is to give a formal definition-by-mathematical-induction.