4 added 3 characters in body; edited body; added 2 characters in body; edited body

I risk reviving a settled matter.

I aim these sketchy remarks at the expert teacher - not at the neophyte student.

A motivation for open sets in topology might begin with a critique of measurement. Though we often think of measurement in continuous terms, practical measurement really always comes down essentially to answering Boolean questions. Thus the real-valued distance function $d(x,y)$ on a metric space carries the same information as a Boolean-valued function $D(x,y,r)$ where $D(x,y,r)=1$ iff $d(x,y)\leq r$. The Dedekind construction of the reals reflects the sort of divide-and-conquer process of actual measurement, the sort of process that takes us from $D$ to $d$, but not in finite terms.

Now in a non-metric topological space you just allow yourself a richer set of questions than you can index with a variable $r$ running over the reals.

Indeed mathematicians often identity sets with properties - having such and such a property means belonging to the (sub)set of all elements (of a given set) that have that property. Then we can measure the proximity of two things by which properties (that we care about) they share.

At a technical level, the previous paragraph has this reflection: just as a metric allows you to embed a metric space in a product of copies of the positive reals, a topology allows you to embed a general space in a product of copies of the $2$-element Sierpiński space.

Now a student might reasonalby reasonably challenge the appearance of Sierpiński space in this fundamental role: why the asymmetry? why have just one open point? if binary decisions lie at the heart of the story, why not take as fundamental the $2$-element space discrete?

I say the choice of Sierpiński space mirrors an aspect of practical life. For certain questions, observation may, on the negative side, supply a full refutation, but on the positive side only at best lend support and never full confirmation. For example, we may discover after careful observation that two quantities are not equal, but often we can only amass evidence that they are equal pending more precise measurement. Another example, when we witness a demise we learn that something wasn't eternal, but observation can never confirm that something is eternal.

This concept of decidability motives the axiomatic closure properties of open sets. Intuitively, an open property admits confirmation by a finite amount of evidence. An arbitrary disjunction of open properties (a union) gets confirmed by confirming any one of them and thus also requires only a finite amount of evidence. But a conjunction of open properties requires confirming them all, so we must limit ourselves, a at least a priori, to finite conjunctions (intersections).

In summary, the Sierpiński space may seem like a curiosity, in if not a monstrosity, but it captures the essence of topology. We have open sets because we care about continuous maps to Sierpiński space, whether consciously or not. We care about continuous maps to Sierpiński space because we care about properties whether they are decidable or not (in the sense of the intuitionists, not in the sense of Turing), i.e., whether or not they disconnect the universe of possibilities. Accepting Sierpiński space commits you to accepting the subspaces of its self-products. The real conceptual hurdle for the topological neophyte lies in contemplating the ubiquity of undecidable properties.

Grothendieck topologies fit very nicely into this point of view (as it leads to topos theory). In essence, Grothendieck challenges the doctrine of identifying properties with subsets. For Grothendieck, a given property may only become visible if one breaks a symmetry or observes some distinction between objects that previously seemed identical. Thus singling out a property might require taking a cover rather than only passing to a subset.

3 added 452 characters in body

I risk reviving a settled matter.

I aim these sketchy remarks at the expert teacher - not at the neophyte student.

A motivation for open sets in topology might begin with a critique of measurement. Though we often think of measurement in continuous terms, practical measurement really always comes down essentially to answering Boolean questions. Thus the real-valued distance function $d(x,y)$ on a metric space carries the same information as a Boolean-valued function $D(x,y,r)$ where $D(x,y,r)=1$ iff $d(x,y)\leq r$. The Dedekind construction of the reals reflects the sort of divide-and-conquer process of actual measurement, the sort of process that takes us from $D$ to $d$, but not in finite terms.

Now in a non-metric topological space you just allow yourself a richer set of questions than you can index with a variable $r$ running over the reals.

Indeed mathematicians often identity sets with properties - having such and such a property means belonging to the (sub)set of all elements (of a given set) that have that property. Then we can measure the proximity of two things which properties (that we care about) they share.

At a technical level, the previous paragraph has this reflection: just as a metric allows you to embed a metric space in a product of copies of the positive reals, a topology allows you to embed a general space in a product of copies of the $2$-element Sierpiński space.

Now a student might reasonalby challenge the appearance of Sierpiński space in this fundamental role: why the asymmetry? why have just one open point? if binary decisions lie at the heart of the story, why not take as fundamental the $2$-element space discrete?

I say the choice of Sierpiński space mirrors an aspect of practical life. For certain questions, observation may, on the negative side, supply a full refutation, but on the positive side only at best lend support and never full confirmation. For example, we may discover after careful observation that two quantities are not equal, but often we can only amass evidence that they are equal pending more precise measurement. Another example, when we witness a demise we learn that something wasn't eternal, but observation can never confirm that something is eternal.

This concept of decidability motives the axiomatic closure properties of open sets. Intuitively, an open property admits confirmation by a finite amount of evidence. An arbitrary disjunction of open properties (a union) gets confirmed by confirming any one of them and thus also requires only a finite amount of evidence. But a conjunction of open properties requires confirming them all, so we must limit ourselves, a least a priori to finite conjunctions (intersections).

In summary, the Sierpiński space may seem like a curiosity, in not a monstrosity, but it captures the essence of topology. We have open sets because we care about continuous maps to Sierpiński space, whether consciously or not. We care about continuous maps to Sierpiński space because we care about properties whether they are decidable or not (in the sense of the intuitionists, not in the sense of Turing), i.e., whether or not they disconnect the universe of possibilities. Accepting Sierpiński space commits you to accepting the subspaces of its self-products. The real conceptual hurdle for the topological neophyte lies in contemplating the ubiquity of undecidable properties.

Grothendieck topologies fit very nicely into this point of view (as it leads to topos theory). In essence, Grothendieck challenges the doctrine of identifying properties with subsets. For Grothendieck, a given property may only become visible if one breaks a symmetry or observes some distinction between objects that previously seemed identical. Thus singling out a property might require taking a cover rather than only passing to a subset.

2 added 565 characters in body

I risk reviving a settled matter.

I aim these sketchy remarks at the expert teacher - not at the neophyte student.

A motivation for open sets in topology might begin with a critique of measurement. Though we often think of measurement in continuous terms, practical measurement really always comes down essentially to answering Boolean questions. Thus the real-valued distance function $d(x,y)$ on a metric space carries the same information as a Boolean-valued function $D(x,y,r)$ where $D(x,y,r)=1$ iff $d(x,y)\leq r$. The Dedekind construction of th the reals reflects the sort of divide-and-conquer process of actual measurement, the sort of process that takes us from $D$ to $d$, but not in finite terms.

Now in a non-metric topological space you just allow yourself a richer set of questions than you can index with a variable $r$ running over the reals.

Indeed mathematicians often identity sets with properties - having such and such a property means belonging to the (sub)set of all elements (of a given set) that have that property. Then we can measure the proximity of two things which properties (that we care about) they share.

At a technical level, the previous paragraph has this reflection: just as a metric allows you to embed a metric space in a product of copies of the positive reals, a topology allows you to embed a general space in a product of copies of the $2$-element Sierpiński space.

Now a student might reasonalby challenge the appearance of Sierpiński space in this fundamental role: why the asymmetry? why have just one open point? if binary decisions lie at the heart of the story, why not take as fundamental the $2$-element space discrete?

I say the choice of Sierpiński space mirrors an aspect of practical life. For certain questions, observation may, on the negative side, supply a full refutation, but on the positive side only at best lend support and never full confirmation. For example, we may discover after careful observation that two quantities are not equal, but often we can only amass evidence that they are equal pending more precise measurement. Another example, when we witness a demise we learn that something wasn't eternal, but observation can never confirm that something is eternal.

This concept of decidability motives the axiomatic closure properties of open sets. Intuitively, an open property admits confirmation by a finite amount of evidence. An arbitrary disjunction of open properties (a union) gets confirmed by confirming any one of them and thus also requires only a finite amount of evidence. But a conjunction of open properties requires confirming them all, so we must limit ourselves, a least a priori to finite conjunctions (intersections).

In summary, the Sierpiński space may seem like a curiosity, in not a monstrosity, but it captures the essence of topology. We have open sets because we care about continuous maps to Sierpiński space, whether consciously or not. We care about continuous maps to Sierpiński space because we care about properties whether they are decidable or not (in the sense of the intuitionists, not in the sense of Turing), i.e., whether or not they disconnect the universe of possibilities. Accepting Sierpiński space commits you to accepting the subspaces of its self-products. The real conceptual hurdle for the topological neophyte lies in contemplating the ubiquity of undecidable properties.

1