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I am studying topology, and trying to get some intuition for why the typical open set formulation (closed under arbitrary union, finite intersection) was selected as the definition.

RougeSegwayUser
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  • The answers to a similar question are quite relevant: http://math.stackexchange.com/questions/15007/topological-space-definition-in-terms-of-open-sets. T – lhf May 29 '13 at 13:06
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    You might get something out of http://math.stackexchange.com/questions/70445/origins-of-the-modern-definition-of-topology? – Gerry Myerson May 29 '13 at 13:07
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    Perhaps this might also help: http://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets-closed – Dejan Govc May 29 '13 at 13:46
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    Your question is slightly ambiguous. Interpretation (1): There are several equivalent ways to define the notion of topological space; why is this the one most often used? Interpretation (2): Irrespective of which of the equivalent definitions is used, why is this particular mathematical structure worth singling out? Can you clarify which you had in mind? – Brian M. Scott May 29 '13 at 19:40
  • Sorry, you'ure right, it's ambigiuous .I meant the first one. – RougeSegwayUser May 29 '13 at 21:18
  • It's an abstraction of the properties of open sets in metric spaces. You really should study analysis before topology. Then the abstraction becomes kind of obvious. – John Douma Jul 30 '21 at 20:13

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Unfortunately I don't know the precise origin of the definition of a topology as a collection of "open" sets satisfying the standard axioms.

Nevertheless, one should note that it is pretty easy to move between the open set definition and the closed set definition (just taking complements), so it's probably not worth putting too much weight on which of the two structures is singled out in the standard definition.

You might be interested to know that in some contexts, the topological structure is naturally defined in terms of closed sets rather than open sets. For example, the Zariski topology on algebraic sets over a field, or on the Spec of a (commutative, with unity) ring, is naturally defined in terms of the closed sets. The closed sets are defined directly in terms of ideals, while a general open set doesn't really have a direct description except as the complement of some closed set; only the distinguished opens admit a direct description (and while these form a base for the topology, but not every open set need be distinguished).

Matt E
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