In the product topology, the closure of a product of subsets is equal to the product of the closures.
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Anonymous2013-09-01 0:55
The observation here is that one can define a function to select from an infinite number of pairs of shoes by stating for example, to choose the left shoe. Without the axiom of choice, one cannot assert that such a function exists for pairs of socks, because left and right socks are (presumably) indistinguishable from each other.
A (covariant) functor F from a category C to a category D, written F : C → D, consists of:
for each object x in C, an object F(x) in D; and
for each morphism f : x → y in C, a morphism F(f) : F(x) → F(y),
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Anonymous2013-09-01 2:26
Isomorphism: f : X → Y is called an isomorphism if there exists a morphism g : Y → X such that f ∘ g = idY and g ∘ f = idX. If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so f is an isomorphism, and g is called simply the inverse of f. Inverse morphisms, if they exist, are unique. The inverse g is also an isomorphism with inverse f. Two objects with an isomorphism between them are said to be isomorphic or equivalent.
Representable functors: We can generalize the previous example to any category C. To every pair X, Y of objects in C one can assign the set Hom(X,Y) of morphisms from X to Y. This defines a functor to Set which is contravariant in the first argument and covariant in the second, i.e. it is a functor Cop × C → Set. If f : X1 → X2 and g : Y1 → Y2 are morphisms in C, then the group homomorphism Hom(f,g) : Hom(X2,Y1) → Hom(X1,Y2) is given by φ ↦ g o φ o f.