I'm new to Topology and need a little help in working out a general intuition of how to build proofs. I have a question that is asking me to show that a quotient map (from a topology onto it's quotient topology, defined by an equivalence relation) is a local homeomorphism.
I know the definition of a local homeomorphism and I realise that I need to somehow show that there exists a neighbourhood of each point in the topology that when the quotient map is restricted to it it becomes a local homeomorphism, but what would constitute a complete proof that that is the case?
Asre there any good examples or templates for questions like these that anyone can recommend?
I can post the question if needs be, but I don't want an answer, just an explanation of what needs to be done for the proof to be complete.