## A General Topology WorkbookThis book has been called a Workbook to make it clear from the start that it is not a conventional textbook. Conventional textbooks proceed by giving in each section or chapter first the definitions of the terms to be used, the concepts they are to work with, then some theorems involving these terms (complete with proofs) and finally some examples and exercises to test the readers' understanding of the definitions and the theorems. Readers of this book will indeed find all the conventional constituents--definitions, theorems, proofs, examples and exercises but not in the conventional arrangement. In the first part of the book will be found a quick review of the basic definitions of general topology interspersed with a large num ber of exercises, some of which are also described as theorems. (The use of the word Theorem is not intended as an indication of difficulty but of importance and usefulness. ) The exercises are deliberately not "graded"-after all the problems we meet in mathematical "real life" do not come in order of difficulty; some of them are very simple illustrative examples; others are in the nature of tutorial problems for a conven tional course, while others are quite difficult results. No solutions of the exercises, no proofs of the theorems are included in the first part of the book-this is a Workbook and readers are invited to try their hand at solving the problems and proving the theorems for themselves. |

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### Contents

TOPOLOGICAL SPACES | 3 |

MAPPINGS OF TOPOLOGICAL SPACES | 19 |

INDUCED AND COINDUCED TOPOLOGIES | 23 |

CONVERGENCE | 31 |

SEPARATION AXIOMS | 43 |

COMPACTNESS | 57 |

CONNECTEDNESS | 65 |

ANSWERS | 71 |

ANSWERS FOR CHAPTER 2 | 91 |

ANSWERS FOR CHAPTER 3 | 95 |

ANSWERS FOR CHAPTER 4 | 103 |

ANSWERS FOR CHAPTER 5 | 113 |

ANSWERS FOR CHAPTER 6 | 131 |

ANSWERS FOR CHAPTER 7 | 141 |

FURTHER READING | 149 |

150 | |

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### Common terms and phrases

adherent point ANSWERS FOR CHAPTER belongs CE(F closed sets closure completely normal connected component continuous mapping contradiction countable dense subset countably compact define digital topology disconnected discrete topology disjoint T-open sets distinct points dyadic rationals element equivalence relation eventually Exercise f a mapping f is continuous family of sets filter F finite family finite intersection property finite subcover finite subset follows Hausdorff space includes a set integer Let E,T Let F limit point locally connected mapping f natural numbers open cover open set p a point particular point topology point of f positive real number Prove R-class rational numbers relative to F saturated separated subsets sequence sequentially compact set in F set which includes subspace topology Suppose f T-closed set T-closed subset T-open set containing T-open subset T')-continuous Theorem topological space topology induced trivial topology ultrafilter ultranet union w-accumulation point