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## QuoteRef: simmGF_1963

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ThesaHelp:
references sa-sz
Group:
sets
Topic:
universal data type
Topic:
topology
Topic:
real numbers and floating point numbers
Topic:
object and value equivalence
Topic:
ordered data types
Topic:
lattice theory of types
Group:
mathematics

#### Reference

Simmons, G.F., "Introduction to topology and modern analysis", New York, pp. McGraw S-Hill Book Company, 1963. Google

Quotations
 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: Schroeder-Bernstein theorem--if 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 choice--given a non-empty class of disjoint non-empty 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 one-to-one, 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

Related Topics

ThesaHelp: references sa-sz (237 items)
Group: sets   (7 topics, 148 quotes)
Topic: universal data type (18 items)
Topic: topology (29 items)
Topic: real numbers and floating point numbers (37 items)
Topic: object and value equivalence (60 items)
Topic: ordered data types (8 items)
Topic: lattice theory of types (15 items)
Group: mathematics   (23 topics, 554 quotes)

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