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69399 37510 58209 74944 59230 78164 06286 20899

and so far I know pi to 80 decimal places

3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899.

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3.14159 26535 89793 23846 26433 83279 50288 41971

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3.14159 26535 89793 23846 26433 83279

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Let *C* be a category with objects ob(*C*), let *X* be a topological space, and define * Top*(

i) for every open subset *U* of *X*, an object * F*(

ii) for every inclusion of an open subset *V* of *X* in an open subset *U* of *X*, an arrow p_{UV}:* F*(U)->

condition 1) p_{UU} is the identity arrow of * F*(U),

condition 2) p* _{UW}*=p

condition 3) for any open covering {*V _{i}*} of any open subset

condition 4) for any open covering {*V _{i}*} of any open subset

**Presheaves**

The definition of a presheaf is weaker than that of a sheaf. A *preasheaf* on *X* consists of

i) for every open subset *U* of *X*, an object * F*(

ii) for every inclusion of an open subset *V* of *X* in an open subset *U* of *X*, an arrow p_{UV}:* F*(U)->

condition 1) p_{UU} is the identity arrow of * F*(U) and

condition 2) p* _{UW}*=p

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Let **k** be an algebraically closed field.* Affine n-space *over **k** is the set of all sets (a_{1},…,a_{n}) of n elements of **k***, *and is denoted by* A^{n}_{k} *or just

Suppose *f* is a member of the polynomial ring A in n variables over **k** and **P**=(a_{1},…,a_{n}) is a point in **A ^{n}. **Then

Let T be some subset of A. The set Z(T) of zeros of T is defined to be the greatest subset of elements **P** of **A ^{n}** such that

A subset of a topological space is *irreducible* if it is nonempty and contains no two proper closed subsets of which it can be written as the union.

*Projective n-space* **P ^{n}** over an algebraically closed field is defined as the set of equivalence classes of elements of

**Varieties**

*Varieties* fall into four types, as detailed below.

Let **A ^{n} **have the Zariski topology. The closed irreducible subsets of

Let T be some subset of the polynomial ring in n+1 variables over **k**. Z(T) is then defined to be be the set of all equivalence classes in **P ^{n}** such that

[Note: there also exists the (obviously) less concrete notion of an *abstract variety*.]

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