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mjordan2nd

- 177

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__Geometry, Topology and Physics__, the following definition of a topological space is given.

Let [itex]X[/itex] be any set and [itex]T=\{U_i | i \in I\}[/itex] denote a certain collection of subsets of [itex]X[/itex]. The pair [itex](X,T)[/itex] is a topological space if [itex]T[/itex] satisfies the following requirements

1.) [itex]\emptyset, X \in T[/itex]

2.) If [itex]T[/itex] is any (maybe infinite) subcollection of [itex]I[/itex], the family [itex]\{U_j|j \in J \} [/itex] satisfies [itex]\cup_{j \in J} U_j \in T[/itex]

3.) If K is any finite subcollection of [itex]I[/itex], the family [itex]\{U_k|k \in K \} [/itex] satisfies [itex]\cap_{k \in K} U_k \in T[/itex]

I'm not very familiar with mathematical notation and conventions, so this is a little confusing to me. Presently I have a few questions:

1.) What do [itex]I[/itex], [itex]J[/itex], and [itex]K[/itex] denote? I don't think they have been referenced before in the text. Do they just represent integers?

2.) I'm a little confused by the first criterion in the definition. I thought [itex]T[/itex] is a collection of subsets of [itex]X[/itex]. Does the first criterion mean that the set [itex]X[/itex] is in [itex]T[/itex] or that a union of sets in [itex]T[/itex] should yield [itex]X[/itex]?

3.) Why are the last two criterion not considered self evident from the definition of [itex]T[/itex]. For instance, for the second criterion if [itex]J<I[/itex] how can a union of [itex]U_j[/itex] be anything but within [itex]T[/itex]?

Thanks.