Analogy between quotient groups and quotient topologyI am trying to see "analogy" between the two concepts in the title, although I am familiar with independent definitions of them.(1) Let G be a group, N a normal subgroup. Define an equivalence relation on G by g 1 ∼ g 2 if g 1 g 2 − 1 ∈ N. Let [ g 1 ] denote equivalence class of g, and G/N the set of equivalence classes. Then we have a canonical map φ : G → G / N, where domain is group, and codomain is "set" right now.Then the quotient structure on G/N is a "group structure" such that the map φ becomes a homomorphism. Such a structure is unique. We call G/N quotient group.Step 2(2) Let X be a topological space, ∼ an equivalence relation on X, and [x] denote equivalence class. Let X/∼ denote the set of equivalence classes. Then there is canonical map φ : X → X / ∼ , where domain is topological space and codomain is just set right now.The quotient topology on X / ∼ is a topology such that the map φ is continuous.But, I was feeling that there could be more than one topological structures on X/∼ which make the canonical map continuous. For example, take nice example of quotient topology on one hand, and structure of indiscrete topology on X/∼ on the other hand.Question: What is exact definition of quotient topology which makes it unique in some sense? (compare with quotient structure in (1))

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Analogy between quotient groups and quotient topologyI am trying to see &quot;analogy&quot; between the two concepts in the title, although I am familiar with independent definitions of them.(1) Let G be a group, N a normal subgroup. Define an equivalence relation on G by       g    1    ∼      g    2   if       g    1        g    2          −      1        ∈  N. Let   [      g    1    ] denote equivalence class of g, and G/N the set of equivalence classes. Then we have a canonical map   φ  :  G  →  G      /    N, where domain is group, and codomain is &quot;set&quot; right now.Then the quotient structure on G/N is a &quot;group structure&quot; such that the map φ becomes a homomorphism. Such a structure is unique. We call G/N quotient group.Step 2(2) Let X be a topological space, ∼ an equivalence relation on X, and [x] denote equivalence class. Let X/∼ denote the set of equivalence classes. Then there is canonical map   φ  :  X  →  X      /        ∼  , where domain is topological space and codomain is just set right now.The quotient topology on   X      /        ∼   is a topology such that the map   φ is continuous.But, I was feeling that there could be more than one topological structures on X/∼ which make the canonical map continuous. For example, take nice example of quotient topology on one hand, and structure of indiscrete topology on X/∼ on the other hand.Question: What is exact definition of quotient topology which makes it unique in some sense? (compare with quotient structure in (1))

Phillip Fletcher · Accepted Answer

Step 1The quotient topology is the unique topology on the set   X      /        ∼   of equivalence classes such that for any topological space Y and any map   f  :  X      /        ∼    →  Y, f is continuous iff   φ  ∘  f is continuous. (Note that in the case where you take   Y  =  X      /        ∼   and f to be the identity map, this says in particular that   φ itself is continuous.) The analogous characterization also works for the quotient group structure, and pretty much any other quotient structure you&#039;ll ever run into.Step 2Another way to phrase this in the case of topological spaces is that the quotient topology is the finest topology that makes   φ continuous. Indeed, for any topology that makes   φ continuous,   φ  ∘  f will be continuous whenever f is continuous. So assuming   φ is continuous, the condition above is just that f is continuous whenever   φ  ∘  f is: that is, there are as many continuous maps out of   X      /        ∼   as possible (given the condition that   φ is continuous). More continuous maps out of a space means a finer topology, so this is saying it&#039;s the finest topology for which   φ is continuous.

Oscar Burton · Accepted Answer

Step 1Both these quotients have the following universal property: Let X be a group (topological space). In the case of groups, you can associate a equivalence relation to every normal subgroup, which is   x  ∼  y if they belong to the same coset. Then G/N is just G/∼ as a set.So now we just view the common problem of going mod an equivalence relation. The quotient structure on X/∼ is one that makes the quotient map   p  :  X  →  X      /    ∼ have the universal property that given a map (continous or group homomorphism respectively)   g  :  X  →  Y with   f  (  x  )  =  f  (  y  ) for all   x  ∼  y then there exists a unique             g      ¯        :  X      /    ∼→  Y such that   g  =  p  ∘            g      ¯      .Step 2You can try and workout that in the case of groups, this universal property makes sure that the quotient group structure on G/N is the only one where   G  →  G      /    ∼ is a group homomorphism. But for groups we can&#039;t go mod arbitrary equivalence relations only those arising from normal subgroups.On the other hand for topological spaces, there can be other topological space structures on   X      /    ∼ which make   X  →  X      /    ∼ continuous, but only the quotient map has the universal property.

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