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Milewski E.G. Topology Problem Solver
Milewski E.G.  Topology Problem Solver







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: Topology Problem Solver

: Milewski E.G.

:

Thorough coverage is given to the fundamental concepts of topology, axiomatic set theory, mappings, cardinal numbers, ordinal numbers, metric spaces, topological spaces, separation axioms, Cartesian products, the elements of homotopy theory, and other topics. A comprehensive study aid for the graduate student and beyond.


: en

: /

:

ed2k: ed2k stats

: 1st edition

: 1994

: 744

: 05.04.2008

: | | ID
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$F_\sigma$ and $G_\delta$ sets      1138 1139 1141
$R_n$ space      102
$T_0$spaces      131 132
$T_1$-spaces      1234 133 134 135 136 137
$T_1$space      1234
$T_2$spaces      138 139 1310 1311 1312 1313
$T_3$spaces      1318 1319 1320 1321
$T_4$spaces      1326 1327 1328
Absorption, law      110
Adding of handles      9105
Additive families      419 420 421
Affine geometry      97 98 9
Alexandroff compactification      1650 1651
AND, OR, INVERTER gates      113 114
Aristotelian logic      12
Axiom of Choice      71
Axiomatic formulation      62
Axiomatic formulation, Axiom of choice      326
Axiomatic set theory      224
Baire's category theorem      1839
Basis, properties      1115 1116 1122 1123 1124
Boolean algebra      225
Borel sets      1142 1143 1144
Boundary      1135
Bounded sets and bounded functions      1026
Brower's fixed point theorem      988 989 990
Burali-forti paradox      841
Canonical mapping (projection)      323
Cantor's theorem      1818
Cantor's theorem, Cardinal number of a power set      610 611 612 613
Cardinal numbers addition      614 615 617 618
Cardinal numbers exponentiation      623 624 625 626 627 628
Cardinal numbers product      616 617 618 619 620 621 622
Cardinal numbers, cardinal numbers      54
Cartesian product      35
Cartesian product, countable sets      59
Cartesian product, general cartesian products      428 429 430 431 1412 1413
Cartesian product, invariants      147 148
Cartesian product, metric spaces      106 1416 1417
Cartesian product, projections      141 142 143 144 145 146
Cartesian product, properties      36 37 38
Cauchy space, Cauchy sequence      186 187 188 189 1810 1811 1812 1813 1814 1827 1828
Chain      710
Characteristic function      425
Chromatic numbers, Color problems, chromatic numbers      969 978 979
Chromatic numbers, Color problems, five color theorem      980
Chromatic numbers, Color problems, four color problem      971 981
Chromatic numbers, Color problems, problem of five regions      968
Chromatic numbers, Color problems, regular maps      970 972 974 975 976
Chromatic numbers, Color problems, seven color theorem      977
Chromatic numbers, Color problems, six color theorem      973
Clausius law      16
Closed ball      1014
Closed sets      1014 1015 1016 1029 1030 1031
Closure, closure of A      1032
Closure, closure of a set      1127
Closure, properties      1128 1129 1130
Cluster point, Derived set      1131 1132 1133
Compact spaces      161 162 163 164 165 167
Compact spaces, countable      1626 1627
Compact spaces, locally      1632 1633 1634 1637
Compact spaces, locally compact Hausdorff spaces      1635 1636
Compact spaces, new      1618 1620 1623
Compact spaces, properties      168 169 1611
Compact spaces, sequentially      1624 1625
Compact spaces, sequentially and countable properties      1628 1629 1630 1631
Compactifications      1648 1649
Compactness, compactness in metric spaces      1644 1645 1646 1647
Compactness, compactness of Hausdorff spaces      1612 1613 1614
components      1729 1730
Components theorems      1731 1732 1733
Congruence classes, affine geometry      97 98 99
Congruence classes, congruence      94 95 96
Congruence classes, isometric transformations      93
Congruence classes, projective geometry      910 911
Congruence classes, rigid transformations      91 92
Congruence classes, topology      912 913 914
Connected and disconnected spaces      171 172
Connected and disconnected spaces, connected subsets of $R^n$      179 1713
Connected and disconnected spaces, connected subspaces      1725 1726 1727 1728
Connected and disconnected spaces, locally connected spaces      1739 1740 1742 1743
Connected and disconnected spaces, locally connected spaces, product      1741 1744
Connected and disconnected spaces, product      1734 1735
Connected and disconnected spaces, properties      173 174 175
Connected and disconnected spaces, totally disconnected spaces      1736 1737 1738
Connected networks      955
Connected surfaces, homotopy classes      927
Connected surfaces, rank      928
Connected surfaces, simply connected surfaces      923 924 925 926
Constituents      215 216
Continuity      121
Continuity at a point, continuity of a point      1218 1219 1220
Continuity at a point, sequential continuity      1221
Continuity of maps      1414 1415
Continuity, applications      1215 1216 1217
Continuity, examples of continuous functions      122 123 125
Continuity, properties of continuous functions      124 1211
Continuity, sequential continuity      1221
Continuity, theorems      126 127 1218 129 1210 1212 1213 1214
Continuous functions      1021 166
Continuous functions, examples      122 123 125
Continuous functions, properties      1022 1023 1024 1025 124 1211
Continuum      1745 1746 1747 1748
Continuum Hypothesis      640 641
Contract mapping      1824
Contraposition      16
Convergence      1017 1019
Convex subsets of $R^n$      1718 1720
Coverings of space      1223
Coverings, Partitions      415 416 417 418
Cross-cap      9109 9110
CURVES      915 916 917
d-ball      107 108
DeMorgan's laws      13 14
DeMorgan's Theorem      213 214
Dense and isometric      1833 1834 1836 1837 1838
Dense sets      1136
Denumerable and countable sets, properties      512 513 514 515 516 518 519 520 521
Denumerable sets      54 55 510
Design of circuits      113 114 116
Diagonal      149 1410 1411
Diagonal, inverse relations      314
Diameter of a set      1026 1028
Digital systems, AND, OR, INVERTER, gates      113 114
Digital systems, design of circuits      113 114 116
Digital systems, logic gates and Boolean algebra      112
Digital systems, NOR, NAND, gates      115
Digital systems, TTL ICs      116
Discrete topology      1110
Duns Scotus law      16
Equal sets      22
Equivalence, class      321 322 1914 1915 1916 1917 1918 1920
Equivalence, equivalent bases      1125
Equivalence, equivalent sets      51 52 522 523 524
Equivalence, equivalent statements      15
Equivalence, relation      317 318 319 320
Euclidean space      103
Euclidean topology      113 1112
Euclidean topology, basis      1114 1117
Euler characteristics      9106 9107 9108
Euler characteristics, and plane diagram      9101
Euler characteristics, Euler rule      929 930
Euler characteristics, index      942 943 944 945 946
Euler characteristics, polyhedra      933 934 935
Euler characteristics, regular divisions      931 932
Euler characteristics, relation with genus      939 940
Euler characteristics, sinks and sources      941
Euler characteristics, triangulation method      936 937 938
Euler Rule      929 930
Families of sets      39 419
Family of closed sets      1111
Finite and transfinite numbers      68
Finite intersection property      1610
First countable spaces      153 154 155 156 157 158 159
Five color theorem      980
Five regions, problem      968
Fixed point theorems, Brower's fixed point theorem      988 989 990
Fixed point theorems, puzzles      991
Fixed point theorems, rotation      987
Fixed point theorems, special cases      992 993 994 995
Four color problem      971 981
Functions, $f^{-1}$, properties      413 414
Functions, composition      45 46 47 48
functions, domain      41
Functions, functions separating points      1336 1337
Functions, induced functions      412
Functions, inverse function      44
Fundamental group      1922 1923 1924
Genus      918 919
Graph, equal functions      43
Hausdorff maximality principle      731 732
Homeomorphic functions      1619
Homeomorphic spaces, examples      1235 1238 1239
Homeomorphic spaces, homeomorphism      1230 1231
Homeomorphic spaces, theorems      1233 1234 1236 1237
Homeomorphic spaces, topological properties      1232
Homotopic functions      191 192 193 1917
Homotopic mappings      199
Homotopy as an equivalence relation      194 195 1912
Homotopy class      196 1927
Ideals, filters      221 222
Identification spaces      1240
Identification spaces, examples      1241 1242
Image, inverse image      49 410
Implication      15 16
inclusion      23
Independent sets      217
Induced metric      182 183
Infinite and finite sets      51 53
Interior      1134
Isometric transformations      93
Jordan curve, inside and outside      982
Jordan curve, Jordan curve theorem      983 984 985
Jordan curve, Kuratowski's lemma      737
Jordan curve, puzzles      986
Lattice points      55
Least upper bound and greatest lower bound      718 719 720
Lebesque number of cover      1638
Lexicographic ordering      829 830 831
Limit of sequence      1017 1018 1020
Limit point      1832
Lindeloef spaces      1514 1515 1516
Logic gates and Boolean algebra      112
Maximal and minimal elements      721 722 723 724
Maximal and minimum      725 726 727
Metric spaces, countable      184 1819
Metric spaces, metric space      101 102 104 105 1522 1523 1524 1525 1526
Metric spaces, metric space, complete      1815 1816 1817 1820 1821 1822 1823 1826 1829 1830 1835
Metrizable spaces      181 185 1831
Moebius band and Klein bottle      920 921 922 999
Multiplicative families      419 420 421
Nbdfinite family      1222
Neighborhood, adherent points      1127
Networks, connected networks      955
Networks, planar and nonplanar networks      949 950 951
Networks, problems and puzzles      960 962 953 964 965 966 957
Networks, single path      952 953 954
Networks, tiesets      958
Networks, traversing a network      959 960 961
Networks, trees      956 957
Networks, vertices      947 948
Non-countable spaces      56
NOR, NAND, gates      115
Normal spaces      1322 1323 1324 1325
Numbers card N and card R      61 63 64
Open and closed functions      125
Open and closed functions, example      1227
Open and closed functions, properties      1226 1228 1229
Open covers, refinements      151 152
Open sets      1011 1012 1013
Order isomorphism      81 82 83
Ordered pair      35
Ordered sets      728 729 730
Ordinal numbers      84
Ordinal numbers, ordering      85 86 87 88 89 810
Ordinal numbers, ordinal number $\omega$      811 812
Ordinal numbers, product      828 832
Ordinal numbers, properties      824 825 826 827
Ordinal numbers, sum      814 815 816 817
Partial order      77 79 711 714
Path connected sets      1714 1715 1716 1717
Path connected sets, properties      1721 1722 1723
Paths, arcs and loops are homotopic      197 198 1910 1911 1912 1913 1921
Piecewise definition of maps      1224
Piecewise definition of maps, coverings of space      1223
Piecewise definition of maps, Nbd-finite family      1222
Planar and nonplanar networks      949 950 951
Plane diagrams      996 997 998
Plane diagrams, Euler characteristic and plane diagram      9101
Plane diagrams, Moebius band and Klein bottle      999
Plane diagrams, real projective plane      9100
Plane diagrams, seven color theorem      9102
Plane diagrams, symbolic representation      9103
Polygonally connected sets      1714 1719
Polyhedra      933 934 935
Power sets      28 29 517
Preorder      74 75 76
Principle of transfinite induction      748 749
Product, properties      833 834 835 836 837
Projective geometry      910 911
Quantifiers $\forall$ and $\exists$      32 33 34
Rank      928
Real projective plane      9100
Real-valued functions      424
Regular divisions      931 932
Regular maps      970 972 974 975 976
Regular spaces      1314 1315 1316 1317
Regular spaces, complete      1338 1339
Relation with genus      939 940
Relation-preserving functions      426 427
Relations      313
Relations, composition      315 316
Remarks      838 839 840
Restriction, extension      43
Rigid transformations      91 92
Rings      220
Rotation      987
Russell paradox      223
Second countable spaces      1510 1511 1513
segments      744 745 746 747
Sentence functions of two variables      324 325
Sentences, sum, product, negation      11
Separable spaces      1512 1517 1518 1519 1520 1521
Separated sets      176
Sequences      1017
Sequential continuity      1221
Set $\Phi(X, Y)$      1027
Set of, algebraic numbers      511
Set of, functions      422 423
Set of, operations      24 25 26 27
Set of, rational numbers      55
Set of, real numbers      56
Sets, elements      21
Seven color theorem      977 9102
Shroeder Bernstein theorem, ordering of cardinal numbers      65 66 67 69
Simple chain      1724
Simply connected surfaces      923 924 925 926
Single path      952 953 954
Sinks and sources      941
Six color theorem      973
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