What's the questions asking me to show?

[andlook]

What is a question in topology asking of me when it says show that the topology of S1 and a quotient space agree?

 

[rasmhop]

That the topologies are equal. A topological space is a set X with a topology T. In your case you are probably given a set S1 with a topology T1 (maybe product topology, maybe subspace topology, maybe discrete topology, maybe some other topology) and you are given a quotient map q : Y \to S1 where Y is a topological space and this induces the quotient topology T2 on S1. You are now asked to show that a set is open in (S1,T1) if and only if it is open in (S1,T2).