What's the questions asking me to show?

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In summary, the question is asking you to show that the topologies of S1 and its quotient space, induced by a quotient map from another topological space, are equal. This means that a set is open in S1 under its original topology if and only if it is open in S1 under the quotient topology.
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andlook
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What is a question in topology asking of me when it says show that the topology of S1 and a quotient space agree?
 
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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 [itex]q : Y \to S1[/itex] 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).
 

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