By Bert Mendelson
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N5. For each point and each neighborhood N of a, there exists a neighborhood O of a such that and O is a neighborhood of each of its points. Proof. For , X is a neighborhood of a, thus N1 is true. N2 is trivial and N3 has already been discussed. To prove N4, let N and M be neighborhoods of a. Then N and M contain open balls B(a; δ1) and B(a; δ2) respectively and therefore N ∩ M contains the open ball B(a; δ), where δ = minimum {δ1, δ2}. To prove N5, let N be a neighborhood of a. 5, O = B(a; δ) is a neighborhood of each of its points.
The subset , which consists of all ordered pairs of the form (a, f(a)) is called the graph of f:A → B. The graph Γf of a function f:X → Y is the subset of X × Y consisting of precisely those points (x, y) for which the statement f(x) = y is true. This set is sometimes written This notation, called the set builder notation, is of the general form {z | P(z) }, where P(z) is some statement which may or may not be true of z. The resulting set is the set of all z, in an appropriate universe, for which P(z) is true.
Two equivalence classes are either disjoint or identical. LEMMA Let R be an equivalence relation on a set E and let for . Then π(a) = π(b). Proof. Let . Then aRc and bRc. Suppose so that aRx. cRa by symmetry, so cRx by transitivity. Another application of transitivity yields bRx, so . Thus . Similarly . By the reflexive property is always true, so the equivalence classes are non-empty and disjoint. Let E/R be the set of equivalence classes, then π:E → E/R is an onto function. E/R is sometimes called the quotient of E by the relation R, and π is called the projection.