I've just started a course in Topology, and was wondering why, when defining a topological space (X, T), T is defined as a family of subsets rather than a set of subsets?
Topology: Family of subsets rather than set of subsets?
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$\begingroup$
general-topology
elementary-set-theory
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2It's a pedagogical choice - the word just means "set," but it is called a "family" because there is an internal relationship between the element sets. Often, beginners have a trouble distinguishing between subsets and sets of subsets, and "family" is sometimes used to make the distinction clearer. It doesn't always work. – 2017-02-02
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0It's another term for set of subsets. That's also why $\mathcal{T}$ or $\mathscr{T}$ is used: the type of letter shows the level of abstraction as it were. It signals taht this set has sets as members. Just capitals are used for subsets, small letters for elements etc. – 2017-02-02
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0From what I can tell from the small amount of set theory/ looking up what a family of sets is, there is a difference between a family of sets and a set of subsets. So in this context is there no difference between them, or do we define it as a family to say avoid some paradoxes associated with the set of all sets or because repeated copies of sets are allowed in a topology? – 2017-02-02
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0No, there is no difference whatsoever. Where did you find other info then? references? – 2017-02-02
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0https://en.wikipedia.org/wiki/Family_of_sets – 2017-02-03
1 Answers
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Often when we have a set of sets we refer to it is a collection or family, it's just a matter of convention. All this definition is saying is $T \subset P(X)$.
Note we can also talk about indexed families $(A_i)_{i \in I}$ where $I$ is known as the index set, and each $A_i$ is a set in the family.