The Gluing lemma states that if you have two functions defined continuously over two closed sets such that the function is equal in the intersection, then the union of the functions is continuous.
My problem is that I want to glue together a countably infinite set of homotopies with fixed end points.
Can I assert that this gluing is continuous? If yes, what is a simple proof of this fact?
To give context, I was trying to simplify a proof of the simple connectedness of the sphere. For the simplification to go through, I have to glue together these homotopies defined over a numerable set of segments of a path.
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