Hey Anteater23, If you have a set X and equivalence relation ~, then the projection X onto X/~ is a quotient map. So this is a specific instance of what I’m talking about in my video. In this instance, it’s typical to call X/~ the “quotient space” or identification space. So to recap, the concepts in my video generalize the key things going on in the equivalence class construction. Sort of like how the integers inspired us to invent a abelian groups and commutative rings, the equivalence class construction inspired us to invent the quotient topology. Does that make sense?
@@DrMcCrady I think I do a bit. So is the quotient topology on the codomain just the final topology on the codomain with respect to the surjective map f? And the quotient topology induced by the projection map is the topology on the quotient set to create the quotient space?
Wait so you can define the quotient topology strictly using a topological space and an onto continuous function only? How and by what are you modding out ?
Hi Hossien, thanks for the support! From my video we’d call f a quotient map, and the topology f (and X’s topology) induce on Y is the quotient topology. As an instance of this, say you have a set X with an equivalence relation ~, then the projection of X onto X/~ is a quotient map. In this special case, we would call X/~ the quotient space. So to recap, I think “quotient space” is typically refers to the equivalence class construction you have in mind, and is a particular example of a quotient map and quotient topology. Does that make sense?
liked ur voice liked the video thnxx
How does this all relate to the quotient space? There’s so many variations of definitions for these things that I’m getting really confused.
Hey Anteater23,
If you have a set X and equivalence relation ~, then the projection X onto X/~ is a quotient map. So this is a specific instance of what I’m talking about in my video. In this instance, it’s typical to call X/~ the “quotient space” or identification space. So to recap, the concepts in my video generalize the key things going on in the equivalence class construction. Sort of like how the integers inspired us to invent a abelian groups and commutative rings, the equivalence class construction inspired us to invent the quotient topology. Does that make sense?
Are you a math grad or undergrad at UCI?
@@DrMcCrady I think I do a bit. So is the quotient topology on the codomain just the final topology on the codomain with respect to the surjective map f? And the quotient topology induced by the projection map is the topology on the quotient set to create the quotient space?
Wait so you can define the quotient topology strictly using a topological space and an onto continuous function only? How and by what are you modding out ?
Hi Hossien, thanks for the support! From my video we’d call f a quotient map, and the topology f (and X’s topology) induce on Y is the quotient topology. As an instance of this, say you have a set X with an equivalence relation ~, then the projection of X onto X/~ is a quotient map. In this special case, we would call X/~ the quotient space. So to recap, I think “quotient space” is typically refers to the equivalence class construction you have in mind, and is a particular example of a quotient map and quotient topology. Does that make sense?
@@DrMcCrady yeah totally so when you mod out by the given relation you obtain a new top space whose open sets are just the pull backs from that map?
Also I didn’t look up but are you an algebraist or an analyst, geometer? I’m applying for PhD schools next year.
Just saw, commutative alg right ?
Yeah that’s it!