QuoteRef: simmGF_1963

topics > all references > ThesaHelp: references sa-sz

references sa-sz
universal data type
real numbers and floating point numbers
object and value equivalence
ordered data types
lattice theory of types


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

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 up

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)

Collected barberCB 4/83
Copyright © 2002-2008 by C. Bradford Barber. All rights reserved.
Thesa is a trademark of C. Bradford Barber.