5 ;;Quote: resolve logical difficulties with sets by assuming a fixed, universal set

12 ;;Quote: the universal set is the intersection of an empty class of sets

19 ;;Quote: define the inverse of a function f by its properties on subsets of f's range

21 ;;Quote: every bounded subset of the reals has a least upper bound

29 ;;Quote: SchroederBernstein theoremif two sets are numerically equivalent to a subset of each other, the sets are numerically equivalent

45 ;;Quote: can not prove Zorn's lemma: if every chain of a partially ordered set is bounded then the set has a maximal element

46 ;;Quote: axiom of choicegiven a nonempty class of disjoint nonempty sets, can form a set with one element from each set

46+;;Quote: Zorn's lemma is equivalent to the axiom of choice

46 ;;Quote: a lattice is a partially ordered set in which every pair has a greatest lower bound and least upper bound; meet and join

60 ;;Quote: a subset of a metric space is open if every member belongs to an open sphere in the set

61 ;;Quote: open sets are closed under unions and finite intersections

65 ;;Quote: x is a limit point of a subset A of a metric space if every open sphere about x includes another element of A

65 ;;Quote: a subset of a metric space is closed if it contains all of its limit points; i.e., separated from its complement

76 ;;Quote: a mapping is continuous iff its inverse maps open sets to open sets

91 ;;Quote: since metrics not needed for defining continuity, suggests defining open sets independently of metrics

92 ;;Quote: a class of subsets is a topology if it is closed under arbitrary unions and finite intersections

93 ;;Quote: the discrete topology of a set is the class of all of its subsets; it is the strongest possible topology

93 ;;Quote: a map between topological spaces is continuous (open) if its inverse (it) maps open sets to open sets

93 ;;Quote: a homeomorphism is a onetoone, continuous open map between topological spaces

94 ;;Quote: a topological property is a property that is preserved under homeomorphisms

95 ;;Quote: a closed set of a topological space is a set whose complement is open; preserved under intersection and finite unions

96 ;;Quote: the closure of a set is the intersection of all closed supersets; only closed sets preserved under closure

96 ;;Quote: a subset is dense in a topological space X if its closure is X

96 ;;Quote: a topological space is separable if it contains a countable, dense subset

96 ;;Quote: a neighborhood of a point is an open set that contains the point

98 ;;Quote: there are many different ways of defining a topological space; the open set approach appears to be the simplest

129 ;;Quote: separation properties of a topology are important because the supply of open sets is tied to the supply of continuous functions

130 ;;Quote: a topological space is T1 if every pair of distinct points separated by a neighborhood; implies points are closed

130 ;;Quote: a topological space is Hausdorff if every pair of distinct points have disjoint neighborhoods

131 ;;Quote: every compact subspace of a Hausdorff space is closed

132 ;;Quote: if continuous functions separate points, the space is Hausdorff

278 ;;Quote: if Tx = c x then x is an eigenvector and c is an eigenvalue
