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Dmitri Pavlov
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This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies.

By a “base” in this answer I mean what appears to be the most common definition: a collection of subsets of a fixed set $A$ such that any finite intersection of elements in the base is a union of elements in the base.

There are also multiple constructions of a site (i.e., a category with a coverage) from a base of a topological space $A$. One can either (A) take the category of all open subsets of $A$, or (B) the category whose objects are open subsets of $A$ that belong to the base. For covering families of some object $V$, one can either take (a) those open covers of $V$ whose elements belong to the base, or (b) open covers of $V$ whose elements are given by the intersection of $V$ and some element of the base. Altogether, there are fourthree different options: A-a, A-b, B-a, B-b, and only option A-b produces a pretopology in the sense of SGA 4.

Axiom PT0 for pretopologies says that any morphism in a covering family admits base changes. Such base changes are always given by the corresponding intersection, provided that the intersection belongs to the category. Thus, PT0 is satsfiedsatisfied for optionoptions A-a, A-b and not satisfied for option B-a. If the base is closed under intersections, then PT0 is also satisfied for option B-a.

Axiom PT1 says that base changes of covering families are covering families. In our case, the base change is given by the intersection with some open subset $V$. Thus, PT1 is satisfied for option A-b and not satisfied for optionoptions A-a, B-a. If the base is closed under intersections, then PT1 is also satisfied for options A-a, B-a, B-b.

Axiom PT2 says that covering families can be composed. This is trivially true for the case under consideration.

Axiom PT3 says that the singleton family consisting of the identity map is a covering family, which is tautologically true in our case.

Thus, option A-b always gives a pretopology, whereas options A-a, B-a, B-b give a pretopology if and only if the base is closed under intersections of pairs.

In particular, open balls in a metric space as a coverage on the category of open subsets trivially form a pretopology using option A-b, and do not for a pretopology in any other optionoptions A-a, B-a.

This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies.

By a “base” in this answer I mean what appears to be the most common definition: a collection of subsets of a fixed set $A$ such that any finite intersection of elements in the base is a union of elements in the base.

There are also multiple constructions of a site (i.e., a category with a coverage) from a base of a topological space $A$. One can either (A) take the category of all open subsets of $A$, or (B) the category whose objects are open subsets of $A$ that belong to the base. For covering families of some object $V$, one can either take (a) those open covers of $V$ whose elements belong to the base, or (b) open covers of $V$ whose elements are given by the intersection of $V$ and some element of the base. Altogether, there are four different options: A-a, A-b, B-a, B-b, and only option A-b produces a pretopology in the sense of SGA 4.

Axiom PT0 for pretopologies says that any morphism in a covering family admits base changes. Such base changes are always given by the corresponding intersection, provided that the intersection belongs to the category. Thus, PT0 is satsfied for option A and not satisfied for option B. If the base is closed under intersections, then PT0 is also satisfied for option B.

Axiom PT1 says that base changes of covering families are covering families. In our case, the base change is given by the intersection with some open subset $V$. Thus, PT1 is satisfied for option A-b and not satisfied for option A-a. If the base is closed under intersections, then PT1 is also satisfied for options A-a, B-a, B-b.

Axiom PT2 says that covering families can be composed. This is trivially true for the case under consideration.

Axiom PT3 says that the singleton family consisting of the identity map is a covering family, which is tautologically true in our case.

Thus, option A-b always gives a pretopology, whereas options A-a, B-a, B-b give a pretopology if and only if the base is closed under intersections of pairs.

In particular, open balls in a metric space as a coverage on the category of open subsets trivially form a pretopology using option A-b, and do not for a pretopology in any other option.

This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies.

By a “base” in this answer I mean what appears to be the most common definition: a collection of subsets of a fixed set $A$ such that any finite intersection of elements in the base is a union of elements in the base.

There are also multiple constructions of a site (i.e., a category with a coverage) from a base of a topological space $A$. One can either (A) take the category of all open subsets of $A$, or (B) the category whose objects are open subsets of $A$ that belong to the base. For covering families of some object $V$, one can either take (a) those open covers of $V$ whose elements belong to the base, or (b) open covers of $V$ whose elements are given by the intersection of $V$ and some element of the base. Altogether, there are three different options: A-a, A-b, B-a, and only option A-b produces a pretopology in the sense of SGA 4.

Axiom PT0 for pretopologies says that any morphism in a covering family admits base changes. Such base changes are always given by the corresponding intersection, provided that the intersection belongs to the category. Thus, PT0 is satisfied for options A-a, A-b and not satisfied for option B-a. If the base is closed under intersections, then PT0 is also satisfied for option B-a.

Axiom PT1 says that base changes of covering families are covering families. In our case, the base change is given by the intersection with some open subset $V$. Thus, PT1 is satisfied for option A-b and not satisfied for options A-a, B-a. If the base is closed under intersections, then PT1 is also satisfied for options A-a, B-a.

Axiom PT2 says that covering families can be composed. This is trivially true for the case under consideration.

Axiom PT3 says that the singleton family consisting of the identity map is a covering family, which is tautologically true in our case.

Thus, option A-b always gives a pretopology, whereas options A-a, B-a give a pretopology if and only if the base is closed under intersections of pairs.

In particular, open balls in a metric space as a coverage on the category of open subsets trivially form a pretopology using option A-b, and do not for a pretopology in options A-a, B-a.

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Dmitri Pavlov
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This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies. By

By a “base” in this answer I mean what appears to be the most common definition: a collection of subsets of a fixed set $A$ such that any finite intersection of elements in the base is a union of elements in the base.

Axiom PT0 for pretopologies says that any morphism inThere are also multiple constructions of a covering family admitssite (i.e., a category with a coverage) from a base changesof a topological space $A$. InOne can either (A) take the category of all open inclusions, cobase changes are given by intersectionssubsets of $A$, so they always exist. or (If one restricts instead toB) the category ofwhose objects are open inclusionssubsets of sets$A$ that belong to the base. For covering families of some object $V$, then cobase changes exist if and only if for any inclusionsone can either take $U→X$(a) those open covers of $V$ whose elements belong to the base, or $V→X$(b) open covers of opens$V$ whose elements are given by the intersection of $V$ and some element of the base. Altogether, there are four different options: A-a, A-b, B-a, B-b, and only option A-b produces a pretopology in the sense of SGA 4.

Axiom PT0 for pretopologies says that any morphism in a covering family admits base changes. Such base changes are always given by the corresponding intersection, provided that the intersection $U\cap V$ also belongs to the category. Thus, PT0 is satsfied for option A and not satisfied for option B. If the base is closed under intersections, then PT0 is also satisfied for option B.)

Axiom PT1 says that pullbacksbase changes of covering families are covering families. This is always true because ofIn our case, the previous axiom andbase change is given by the fact that unions commute with intersection with a fixedsome open subset $V$. Thus, PT1 is satisfied for option A-b and not satisfied for option A-a. If the base is closed under intersections, then PT1 is also satisfied for options A-a, B-a, B-b.

Axiom PT2 says that covering families can be composed. This is trivially true for the case under consideration.

Axiom PT3 says that the singleton family consisting of the identity map is a covering family, which is tautologically true in our case.

Thus, a base foroption A-b always gives a topological space ispretopology, whereas options A-a, B-a, B-b give a pretopology if if and only if itthe base is closed under intersectionintersections of pairs.

In particular, open balls in a metric space as a coverage on the category of open subsets trivially form a pretopology. (If we consider the category of inclusions of open balls themselves without any other open subsets, it does not have cobase changes using option A-b, so open ballsand do not formfor a pretopology in any other option.)

This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies. By a “base” in this answer I mean what appears to be the most common definition: a collection of subsets of a fixed set $A$ such that any finite intersection of elements in the base is a union of elements in the base.

Axiom PT0 for pretopologies says that any morphism in a covering family admits base changes. In the category of open inclusions, cobase changes are given by intersections, so they always exist. (If one restricts instead to the category of open inclusions of sets that belong to the base, then cobase changes exist if and only if for any inclusions $U→X$, $V→X$ of opens in a base, the intersection $U\cap V$ also belongs to the base.)

Axiom PT1 says that pullbacks of covering families are covering families. This is always true because of the previous axiom and the fact that unions commute with intersection with a fixed open subset.

Axiom PT2 says that covering families can be composed. This is trivially true for the case under consideration.

Axiom PT3 says that the singleton family consisting of the identity map is a covering family, which is tautologically true in our case.

Thus, a base for a topological space is a pretopology if and only if it is closed under intersection of pairs.

In particular, open balls in a metric space as a coverage on the category of open subsets trivially form a pretopology. (If we consider the category of inclusions of open balls themselves without any other open subsets, it does not have cobase changes, so open balls do not form a pretopology.)

This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies.

By a “base” in this answer I mean what appears to be the most common definition: a collection of subsets of a fixed set $A$ such that any finite intersection of elements in the base is a union of elements in the base.

There are also multiple constructions of a site (i.e., a category with a coverage) from a base of a topological space $A$. One can either (A) take the category of all open subsets of $A$, or (B) the category whose objects are open subsets of $A$ that belong to the base. For covering families of some object $V$, one can either take (a) those open covers of $V$ whose elements belong to the base, or (b) open covers of $V$ whose elements are given by the intersection of $V$ and some element of the base. Altogether, there are four different options: A-a, A-b, B-a, B-b, and only option A-b produces a pretopology in the sense of SGA 4.

Axiom PT0 for pretopologies says that any morphism in a covering family admits base changes. Such base changes are always given by the corresponding intersection, provided that the intersection belongs to the category. Thus, PT0 is satsfied for option A and not satisfied for option B. If the base is closed under intersections, then PT0 is also satisfied for option B.

Axiom PT1 says that base changes of covering families are covering families. In our case, the base change is given by the intersection with some open subset $V$. Thus, PT1 is satisfied for option A-b and not satisfied for option A-a. If the base is closed under intersections, then PT1 is also satisfied for options A-a, B-a, B-b.

Axiom PT2 says that covering families can be composed. This is trivially true for the case under consideration.

Axiom PT3 says that the singleton family consisting of the identity map is a covering family, which is tautologically true in our case.

Thus, option A-b always gives a pretopology, whereas options A-a, B-a, B-b give a pretopology if and only if the base is closed under intersections of pairs.

In particular, open balls in a metric space as a coverage on the category of open subsets trivially form a pretopology using option A-b, and do not for a pretopology in any other option.

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Dmitri Pavlov
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This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies. By a “base” in this answer I mean what appears to be the most common definition: a collection of subsets of a fixed set $A$ such that any finite intersection of elements in the base is a union of elements in the base.

Axiom PT0 for pretopologies says that any morphism in a covering family admits base changes. This meansIn the category of open inclusions, cobase changes are given by intersections, so they always exist. (If one restricts instead to the category of open inclusions of sets that belong to the base, then cobase changes exist if and only if for any inclusions $U→X$, $V→X$ of opens in a base, the intersection $U\cap V$ also belongs to the base.)

Axiom PT1 says that pullbacks of covering families are covering families. This is always true because of the previous axiom and the fact that unions commute with intersection with a fixed open subset.

Axiom PT2 says that covering families can be composed. This is trivially true for the case under consideration.

Axiom PT3 says that the singleton family consisting of the identity map is a covering family, which is tautologically true in our case.

Thus, a base for a topological space is a pretopology if and only if it is closed under intersection of pairs.

In particular, open balls in a metric space in general do notas a coverage on the category of open subsets trivially form a pretopology, since. (If we consider the intersectioncategory of twoinclusions of open balls isthemselves without any other open subsets, it does not anhave cobase changes, so open ball in generalballs do not form a pretopology.)

This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies. By a “base” in this answer I mean what appears to be the most common definition: a collection of subsets of a fixed set $A$ such that any finite intersection of elements in the base is a union of elements in the base.

Axiom PT0 for pretopologies says that any morphism in a covering family admits base changes. This means that for any inclusions $U→X$, $V→X$ of opens in a base, the intersection $U\cap V$ also belongs to the base.

Axiom PT1 says that pullbacks of covering families are covering families. This is always true because of the previous axiom and the fact that unions commute with intersection with a fixed open subset.

Axiom PT2 says that covering families can be composed. This is trivially true for the case under consideration.

Axiom PT3 says that the singleton family consisting of the identity map is a covering family, which is tautologically true in our case.

Thus, a base for a topological space is a pretopology if and only if it is closed under intersection of pairs.

In particular, open balls in a metric space in general do not form a pretopology, since the intersection of two open balls is not an open ball in general.

This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies. By a “base” in this answer I mean what appears to be the most common definition: a collection of subsets of a fixed set $A$ such that any finite intersection of elements in the base is a union of elements in the base.

Axiom PT0 for pretopologies says that any morphism in a covering family admits base changes. In the category of open inclusions, cobase changes are given by intersections, so they always exist. (If one restricts instead to the category of open inclusions of sets that belong to the base, then cobase changes exist if and only if for any inclusions $U→X$, $V→X$ of opens in a base, the intersection $U\cap V$ also belongs to the base.)

Axiom PT1 says that pullbacks of covering families are covering families. This is always true because of the previous axiom and the fact that unions commute with intersection with a fixed open subset.

Axiom PT2 says that covering families can be composed. This is trivially true for the case under consideration.

Axiom PT3 says that the singleton family consisting of the identity map is a covering family, which is tautologically true in our case.

Thus, a base for a topological space is a pretopology if and only if it is closed under intersection of pairs.

In particular, open balls in a metric space as a coverage on the category of open subsets trivially form a pretopology. (If we consider the category of inclusions of open balls themselves without any other open subsets, it does not have cobase changes, so open balls do not form a pretopology.)

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Dmitri Pavlov
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