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Graph Minor from Wikipedia, the Free Encyclopedia Contents Graph minor From Wikipedia, the free encyclopedia Contents 1 2 × 2 real matrices 1 1.1 Profile ................................................. 1 1.2 Equi-areal mapping .......................................... 2 1.3 Functions of 2 × 2 real matrices .................................... 2 1.4 2 × 2 real matrices as complex numbers ............................... 3 1.5 References ............................................... 4 2 Abelian group 5 2.1 Definition ............................................... 5 2.2 Facts ................................................. 5 2.2.1 Notation ........................................... 5 2.2.2 Multiplication table ...................................... 6 2.3 Examples ............................................... 6 2.4 Historical remarks .......................................... 6 2.5 Properties ............................................... 6 2.6 Finite abelian groups ......................................... 7 2.6.1 Classification ......................................... 7 2.6.2 Automorphisms ....................................... 7 2.7 Infinite abelian groups ........................................ 8 2.7.1 Torsion groups ........................................ 9 2.7.2 Torsion-free and mixed groups ................................ 9 2.7.3 Invariants and classification .................................. 9 2.7.4 Additive groups of rings ................................... 9 2.8 Relation to other mathematical topics ................................. 10 2.9 A note on the typography ....................................... 10 2.10 See also ................................................ 10 2.11 Notes ................................................. 11 2.12 References .............................................. 11 2.13 External links ............................................. 11 3 Associative algebra 12 3.1 Formal definition ........................................... 12 3.1.1 From R-modules ....................................... 12 i ii CONTENTS 3.1.2 From rings .......................................... 13 3.2 Algebra homomorphisms ....................................... 13 3.3 Examples ............................................... 13 3.4 Constructions ............................................. 14 3.5 Associativity and the multiplication mapping ............................. 15 3.6 Coalgebras ............................................... 15 3.7 Representations ............................................ 15 3.7.1 Motivation for a Hopf algebra ................................. 16 3.7.2 Motivation for a Lie algebra .................................. 16 3.8 See also ................................................ 17 3.9 References ............................................... 17 4 Bijection 18 4.1 Definition ............................................... 19 4.2 Examples ............................................... 19 4.2.1 Batting line-up of a baseball team ............................... 19 4.2.2 Seats and students of a classroom ............................... 19 4.3 More mathematical examples and some non-examples ........................ 20 4.4 Inverses ................................................ 20 4.5 Composition .............................................. 20 4.6 Bijections and cardinality ....................................... 20 4.7 Properties ............................................... 21 4.8 Bijections and category theory ..................................... 21 4.9 Generalization to partial functions ................................... 22 4.10 Contrast with ............................................. 22 4.11 See also ................................................ 22 4.12 Notes ................................................. 22 4.13 References ............................................... 23 4.14 External links ............................................. 23 5 Category (mathematics) 24 5.1 Definition ............................................... 25 5.2 History ................................................. 25 5.3 Small and large categories ....................................... 26 5.4 Examples ............................................... 26 5.5 Construction of new categories .................................... 26 5.5.1 Dual category ......................................... 26 5.5.2 Product categories ...................................... 27 5.6 Types of morphisms .......................................... 27 5.7 Types of categories .......................................... 28 5.8 See also ................................................ 28 5.9 Notes ................................................. 28 CONTENTS iii 5.10 References ............................................... 28 6 Complete bipartite graph 30 6.1 Definition ............................................... 30 6.2 Examples ............................................... 30 6.3 Properties ............................................... 31 6.4 See also ................................................ 32 6.5 References ............................................... 32 7 Complete graph 33 7.1 Properties ............................................... 33 7.2 Geometry and topology ........................................ 33 7.3 Examples ............................................... 34 7.4 See also ................................................ 34 7.5 References ............................................... 34 7.6 External links ............................................. 34 8 Complete metric space 35 8.1 Examples ............................................... 35 8.2 Some theorems ............................................ 36 8.3 Completion .............................................. 36 8.4 Topologically complete spaces ..................................... 37 8.5 Alternatives and generalizations .................................... 37 8.6 See also ................................................ 37 8.7 Notes ................................................. 38 8.8 References ............................................... 38 9 Conjugacy class 39 9.1 Definition ............................................... 39 9.2 Examples ............................................... 39 9.3 Properties ............................................... 40 9.4 Conjugacy class equation ....................................... 40 9.4.1 Example ............................................ 41 9.5 Conjugacy of subgroups and general subsets ............................. 41 9.6 Conjugacy as group action ....................................... 41 9.7 Geometric interpretation ....................................... 41 9.8 See also ................................................ 42 9.9 References ............................................... 42 10 Connected space 43 10.1 Formal definition ........................................... 44 10.1.1 Connected components .................................... 44 10.1.2 Disconnected spaces ..................................... 44 iv CONTENTS 10.2 Examples ............................................... 44 10.3 Path connectedness .......................................... 45 10.4 Arc connectedness .......................................... 46 10.5 Local connectedness ......................................... 46 10.6 Set operations ............................................. 46 10.7 Theorems ............................................... 49 10.8 Graphs ................................................. 49 10.9 Stronger forms of connectedness ................................... 50 10.10See also ................................................ 50 10.11References ............................................... 50 10.11.1 Notes ............................................. 50 10.11.2 General references ...................................... 50 11 Connectivity (graph theory) 51 11.1 Connected graph ............................................ 52 11.2 Definitions of components, cuts and connectivity ........................... 52 11.3 Menger’s theorem ........................................... 53 11.4 Computational aspects ......................................... 54 11.5 Examples ............................................... 54 11.6 Bounds on connectivity ........................................ 54 11.7 Other properties ............................................ 54 11.8 See also ................................................ 55 11.9 References ............................................... 55 12 Continuous function 56 12.1 History ................................................ 56 12.2 Real-valued continuous functions ................................... 56 12.2.1 Definition ........................................... 56 12.2.2 Examples ........................................... 59 12.2.3 Non-examples ......................................... 62 12.2.4 Properties ........................................... 63 12.2.5 Directional and semi-continuity ................................ 64 12.3 Continuous functions between metric spaces ............................. 65 12.3.1 Uniform, Hölder and Lipschitz continuity .......................... 66 12.4 Continuous functions between topological spaces ........................... 66 12.4.1 Alternative definitions ..................................... 68 12.4.2 Properties ........................................... 69 12.4.3 Homeomorphisms ....................................... 70 12.4.4 Defining topologies via continuous functions ......................... 70 12.5 Related notions ...........................................
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