FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

ALGEBRA 3
For instance, {a,b}×{0, 1} = {(a, 0), (a, 1), (b, 0), (b, 1)}. Given any arbitrary set J of indices
α, the product over the sets Xα is given by
Xα = {(xα)α∈J| xα ∈ Xα} .
α∈J
Throughout the rest of our present discussion, assume that X, Y and Z are arbitrary
sets. Define Y X = {(yx)x∈X|yx ∈ Y }. This can be thought of both as the set of all X−tuples
of elements of Y (if X is taken to be a set of indices), or as the set of all functions from X
into Y .
Let f be one such function from X into Y . If B ⊂ Y, define f −1 (B) = {x| f(x) ∈ B}.
The set f −1 (B) is called the pullback of B under f.
If X ⊂ Z, and f ′ : Z → Y is such that f(x) = f ′ (x) for every x ∈ X, then f ′ is said to
be an extension of f to Z. Likewise, f is called the restriction of f ′ to X.
The function f is called injective if, for every distinct x1 and x2 in X, f(x1) and f(x2)
are distinct in Y . We say that f is surjective if, for every y in Y , there is an x in X such
that f(x) = y. A function that is both injective and surjective is said to be bijective. If f is
a bijection, then it induces a unique map f −1 : Y → X since the pullback of any point of Y
is nonempty and contains a single point. f −1 is called the inverse of f.
Often, we will need to discuss the size, or cardinality, of a particular set. Let Sn =
{1, 2,...,n}. If there is a bijective function fn : X → Sn for some n ∈ N, then we say that
X is finite, and write |X| = n. If there is a bijective function fN: X → N, then we say that
X is countable, and write |X| = ℵ0. If no such fn or fN exists, we say that X is uncountable.
The following concept will serve us well in the remainder. A sequence in X is a function
f : N → X. Sequences are often written in set notation as {xn| n ∈ N} or {xn}, where
f(n) = xn. A subsequence of a sequence {xn} is the sequence in X formed by restricting
f : N → X to the countable subset N ∗ = {n1,n2,n3,...} of N, where ni < nj if i < j. The
subsequence formed formed by restricting f to N ∗ is denoted {xni }.
Algebra
A group is a set G, together with a binary operation + such that if x,y,z ∈ G: