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... b ∈ B, there is a ∈ A such that f (a) = b. This means that g(b) = g(f (a)) = h(f (a)) = h(b), or g = h, showing that f is epi. On the other hand, suppose f : A → B is epi. Define g, h : B → B/f (A) as follows: g(x) = f (A) and h(x) = x + f (A) for all x ∈ B. Then g(f (a)) = f (A) = f (a) + f (A) = h ...
... b ∈ B, there is a ∈ A such that f (a) = b. This means that g(b) = g(f (a)) = h(f (a)) = h(b), or g = h, showing that f is epi. On the other hand, suppose f : A → B is epi. Define g, h : B → B/f (A) as follows: g(x) = f (A) and h(x) = x + f (A) for all x ∈ B. Then g(f (a)) = f (A) = f (a) + f (A) = h ...
EQUIVALENT NOTIONS OF ∞-TOPOI Seminar on Higher Category
... Proposition 2.4 ([Lur09], Prop 6.2.2.5). Let C be an ∞-category and y : C → PrSh(C) the Yoneda embedding. For an object C in C and a monomorphism i : U → y(C), let (C/C)(i) be the full subcategory of C/C given by those morphisms f : D → C such that y(f ) factors through i. Then the assignment i 7→ ( ...
... Proposition 2.4 ([Lur09], Prop 6.2.2.5). Let C be an ∞-category and y : C → PrSh(C) the Yoneda embedding. For an object C in C and a monomorphism i : U → y(C), let (C/C)(i) be the full subcategory of C/C given by those morphisms f : D → C such that y(f ) factors through i. Then the assignment i 7→ ( ...
MATH1373
... Define a collection T of subsets of R to be of two types: Type1: U , where U is open subset of R. Type 2: R B , where B is closed and bounded subset of R. Show that T is a topology on R . Q14: Let U be a collection of subsets of X . Suppose that and X are in U , and that finite unions and arb ...
... Define a collection T of subsets of R to be of two types: Type1: U , where U is open subset of R. Type 2: R B , where B is closed and bounded subset of R. Show that T is a topology on R . Q14: Let U be a collection of subsets of X . Suppose that and X are in U , and that finite unions and arb ...
Lecture Notes on General Topology
... 1. a ∈ X, butTthere is no open set G(say) in X for which we have (G − {a}) A 6= φ. Hence a is not a limit point of A. 2. b ∈ X, and since the open set containing b are {b, c, d, e} and X, and each contained a point of A different from b. 3. c ∈ X, and since the open set containing c are {c, d}, {a, ...
... 1. a ∈ X, butTthere is no open set G(say) in X for which we have (G − {a}) A 6= φ. Hence a is not a limit point of A. 2. b ∈ X, and since the open set containing b are {b, c, d, e} and X, and each contained a point of A different from b. 3. c ∈ X, and since the open set containing c are {c, d}, {a, ...
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... A topological space is said to be second countable if it has a countable basis. It can be shown that a second countable space is both Lindelöf and separable, although the converses fail. For instance, the lower limit topology on the real line is both Lindelöf and separable, but not second countabl ...
... A topological space is said to be second countable if it has a countable basis. It can be shown that a second countable space is both Lindelöf and separable, although the converses fail. For instance, the lower limit topology on the real line is both Lindelöf and separable, but not second countabl ...