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topologynotes

# topologynotes - AN INTRODUCTION TO TOPOLOGY Fall Semester...

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AN INTRODUCTION TO TOPOLOGY Fall Semester 2007 0. Set Theory In set theory, there are two undefined or primitive concepts: set and is an element of . Taken together, they are the first principals of set theory. Hence their meaning does not derive from other more fundamental concepts, but from their relationship to each other: A set is a collection of objects, each of which is an element of the set. We write x S to denote that x is an element of S . In keeping with common English usage, the word “set” is roughly synonymous with “family,” “collection,” “aggregate,” and “class,” and the phrase “is and element of” is used interchangeably with “is a member of,” “is contained by,” and “belongs to.” Often the members of a set are called “points.” However, we will have many occasions to deal with collections whose members are themselves sets. In this case, it seems inappropriate, though not technically incorrect, to call these sets “points”. Definition. A set A is a subset of a set B , denoted A B if and only if every element of A is also an element of B ; A is a proper subset of B if and only A is a subset of B , but B is not a subset of A . Two sets A and B are equal if and only if A is a subset of B and B is a subset of A . Suppose A is a collection of sets. The union of A , denoted A * or ∪A , is the set to which a point belongs if and only if it belongs to some member of the collection A . The intersection of A , denoted ∩A , is the set to which a point belongs if and only if it belongs to each member of the collection A . If A = { A, B } , then A * and ∩A are conventionally written A B and A B respectively. Similar conventions exist when A is any finite or countably infinite collection. Notation. Sometimes the members of a collection of sets are indexed by another set. If Λ is a set and, for each λ Λ, A λ is a set, then the collection { A λ : λ Λ } is said to be indexed by the set Λ. In this case, { A λ : λ Λ } * may also be denoted by λ Λ A λ or just A λ . Similarly, ∩{ A λ : λ Λ } may also be denoted λ Λ A λ or A λ . Sometimes it is convenient to index the members of a collection. For example, it is natural to index the collection of all sets of the form { 1 , 2 , ..., n } with the natural numbers because of the clear connection between the members of the collection and the members of the indexing set N . Since the set of rational numbers are countable, it is sometimes useful to index them with N and write Q = { r 1 , r 2 , r 3 , ... } . Other times, indexing a set or a collection merely complicates things by introducing superfluous notation. Consequently, one should

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not index the members of a set or a collection without good reason. Using { A i } to denote a collection that may not be countable is unforgivably insidious.
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topologynotes - AN INTRODUCTION TO TOPOLOGY Fall Semester...

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