Topology!!!?

Let p be the map of the real line R onto the three point set A = {a,b,c} defined by p(x)={ a if x%26gt;0, b if x%26lt;0, and c if x=o}. Then it says you can check that the quotient topology on A induced by p is the figure which is a circle around a,b,and c and and a circle around a, and b and a circle around a and a circle around b. So how do you check if this is a quotient topology. I know that you check to see if the pre image of a and b is also open and to check that c is also closed. But I'm not sure how to write it in math language. Any pointers?

Topology!!!?
A map is a quotient map if it is surjective and takes saturated open sets to open sets. Obviously this map is surjective. So what are the saturated open sets? They are {x%26gt;0}, {x%26lt;0}, {x%26gt;0 or x%26lt;0}, empty set, and the entire real line. See if these sets get mapped to open sets.





Alternately you can just check that this is a quotient map directly from the definition- The are only 2^3 = 8 subsets of A, and so it's not difficult to simply go through and check them all to see that they are open if and only if their inverse image under p is open.

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