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This entry is about a generalized notion of topology. For the notion of formal space in the sense of rational homotopy theory, see formal dg-algebra.

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Formal topology

Idea

Formal topology is a programme for doing topology in a finite, predicative, and constructive fashion.

It is a kind of pointless topology; in the context of classical mathematics, it reproduces the theory of locales rather than topological spaces (although of course one can recover topological spaces from locales).

The basic definitions can be motivated by an attempt to study locales entirely through the posites that generate them. However, in order to recover all basic topological notions (particularly those associated with closed rather than open features) predicatively, we need to add a ‘positivity’ relation to the ‘coverage’ relation of sites.

Definitions

A formal topology or formal space is a set $S$ together with

• an element $\top$ of $S$,
• a binary operation $\cap$ on $S$,
• a binary relation $\lhd$ between elements of $S$ and subsets of $S$, and
• a unary relation $\Diamond$ on $S$,

such that

1. $a = b$ whenever $a \lhd \{b\}$ and $b \lhd \{a\}$,
2. $a \lhd U$ whenever $a \in U$,
3. $a \lhd V$ whenever $a \lhd U$ and $x \lhd V$ for all $x \in U$,
4. $a \cap b \lhd U$ whenever $a \lhd U$ or $b \lhd U$,
5. $a \lhd \{ x \cap y \;|\; x \in U,\; y \in V \}$ whenever $a \lhd U$ and $a \lhd V$,
6. $a \lhd \{\top\}$,
7. $\Diamond x$ for some $x \in U$ whenever $\Diamond a$ and $a \lhd U$, and
8. $a \lhd U$ whenever $a \lhd U$ if $\Diamond a$,

for all $a$, $b$, $U$, and $V$.

We interpret the elements of $S$ as basic opens in the formal space. We call $\top$ the entire space and $a \cap b$ the intersection of $a$ and $b$. We say that $a$ is covered by $U$ or that $U$ is a cover of $a$ if $a \lhd U$. We say that $a$ is positive or inhabited if $\Diamond a$. (For a topological space equipped with a strict topological base $S$, taking these intepretations literally does in fact define a formal space; see the Examples.)

Some immediate points to notice:

• If we drop (1), then the hypothesis of (1) defines an equivalence relation on $S$ which is a congruence for $\top$, $\cap$, $\lhd$, and $\Diamond$, so that we may simply pass to the quotient set. In appropriate foundations, we can even allow $S$ to be a preset originally, then use (1) as a definition of equality.
• We can prove that $(S,\cap,\top)$ is a bounded semilattice; if (as the notation suggests) we interpret this as a meet-semilattice, then $a \leq b$ if and only if $a \lhd \{b\}$. Conversely, we could require that $(S,\cap,\top)$ be a semilattice originally, then let (1) say that $a \leq b$ whenever $a \lhd \{b\}$.
• We can prove that $\Diamond a$ holds iff every cover of $a$ is inhabited and that $\Diamond a$ fails iff $a \lhd \empty$. Accordingly, this predicate is uniquely definable (in two equivalent ways, one impredicative and one nonconstructive) in a classical treatment; only in a treatment that is both predicative and constructive do we need to include it in the axioms. See positivity predicate.

Examples

Let $X$ be a topological space, and let $S$ be the collection of open subsets of $X$. Let $\top$ be $X$ itself, and let $a \cap b$ be the literal intersection of $a$ and $b$ for $a, b \in S$. Let $a \lhd U$ if and only $U$ is literally an open cover of $a$, and let $\Diamond a$ if and only if $a$ is literally inhabited. Then $(S,\top,\cap,\lhd,\Diamond)$ is a formal topology.

The above example is impredicative (since the collection of open subsets is generally large), but now let $S$ be a base for the topology of $X$ which is strict in the sense that it is closed under finitary intersection. Let the other definitions be as before. Then $(S,\top,\cap,\lhd,\Diamond)$ is a formal topology.

More generally, let $B$ be a subbase for the topology of $X$, and let $S$ be the free monoid on $B$, that is the set of finite lists of elements of $B$ (so this example is not strictly finitist), modulo the equivalence relation by which two lists are identified if their intersections are equal. Let $\top$ be the empty list, let $a \cap b$ be the concatenation of $a$ and $b$, let $a \lhd U$ if the intersection of $a$ is contained in the union of the intersections of the elements of $U$, and let $\Diamond a$ if the intersection of $a$ is inhabited. Then $(S,\top,\cap,\lhd,\Diamond)$ is a formal topology.

Let $X$ be an accessible locale generated by a posite whose underlying poset $S$ is a (meet)-semilattice. Let $\top$ and $\cap$ be as in the semilattice structure on $S$, and let $a \lhd U$ if $U$ contains a basic cover (in the posite structure on $S$) of $a$. Then we get a formal topology, defining $\Diamond$ in the unique way.

The last example is not predicative, and this is in part why one studies formal topologies instead of sites, if one wishes to be strictly predicative. (It still needs to be motivated that we want $\Diamond$ at all.)

 In dependent type theory

In dependent type theory , if let one doesn’t have a$(\Omega, \mathrm{El}_\Omega)$ be the type of all propositions , then and one let usually the cannot define a relation between a type$\mathrm{<span><del\; class="diffmod">\; S</del><ins\; class="diffmod">\; a</ins></span>}{\in }_{S}A$ S a \in_S A and for the type of all subtypes of$a:S$ S a:S , which and is necessary for defining the coverage$⊲A:S\to \Omega$ \lhd A:S \to \Omega . Instead, as one has to use aTarski universe to define the type of all locally $U$-small subtypes of $S$, and use that to define a locally $U$-small formal topology on $S$.

Let $(U, T)$ be a Tarski universe, and let $\mathrm{Prop}_U$ be the type of all propositions in $U$

$$a \in_S A \coloneqq \mathrm{El}_\Omega(A(a))$$

$$\mathrm{Prop}_U \coloneqq \sum_{A:U} \mathrm{isProp}(T(A))$$

The singleton subtype function is a function $\{-\}:S \to (S \to \Omega)$, such that for all elements $a:S$, $p(a):a \in_S \{a\}$, and for all functions $R:S \to (S \to \Omega)$ such that for all elements $a:S$, $p_R(a):a \in_S R(a)$, the type of functions $(b \in_S \{a\}) \to (b \in_S R(a))$ is contractible for all $a:A$ and $b:A$.

which A comes with anembeddingformal topology or$\mathrm{asType}_U:\mathrm{Prop}_U \hookrightarrow U$formal space . In is addition, a we define the relation$a \in_S^U A$type for$a:S$ a:S S and together with$A:S \to \mathrm{Prop}_U$ as

$$a \in_S^U A \coloneqq \mathrm{isContr}(T(\mathrm{asType}(A(a)))$$

• an element $\top:S$,
• a binary operation $\cap:S \times S \to S$,
• a binary relation $a \lhd A$ between elements of $S$, $a:S$, and subsets of $S$, $A:S \to \Omega$, and
• a unary relation $\Diamond a$ on $S$,

The singleton subtype function is a function $\{-\}:S \to (S \to \mathrm{Prop}_U)$, such that for all elements $a:S$, $p(a):a \in_S^U \{a\}$, and for all functions $R:S \to (S \to \mathrm{Prop}_U)$ such that for all elements $a:S$, $p_R(a):a \in_S^U R(a)$, the type of functions $(b \in_S^U \{a\}) \to (b \in_S^U R(a))$ is contractible for all $a:A$ and $b:A$.

Given a Tarski universe $(U, T)$, a locally $U$-small formal topology or locally $U$-small formal space is a type $S$ together with

• an element $\top:S$,
• a binary operation $\cap:S \times S \to S$,
• a binary relation $a \lhd A$ between elements of $S$, $a:S$, and locally $U$-small subsets of $S$, $A:S \to \mathrm{Prop}_U$, and
• a unary relation $\Diamond a$ on $S$,

such that

1. For all elements $a:S$ and $b:S$, the canonical function
   $$\mathrm{idToFormalTop}(a, b):(a =_S b) \to ((a \lhd \{b\}) \times (b \lhd \{a\}))$$

   is an equivalence of types.

   $$\prod_{a:S} \prod_{b:S} \mathrm{isEquiv}(\mathrm{idToFormalTop}(a, b))$$
2. For all elements $a:S$ and subsets $A:S\to {\mathrm{Prop}}_U\Omega$ A:S \to \mathrm{Prop}_U \Omega, if $a{\in }_S^U{\in }_{S}A$ a \in_S^U \in_S A, then $a \lhd A$
   $$\prod _{a:S}\prod _{A:S\to {\mathrm{Prop}}_U\Omega }(a{\in }_S^U{\in }_{S}A)\to (a⊲A)$$ \prod_{a:S} \prod_{A:S \to \mathrm{Prop}_U} \Omega} (a \in_S^U \in_S A) \to (a \lhd A)
3. For all elements $a:S$ and subsets $A:S\to {\mathrm{Prop}}_U\Omega$ A:S \to \mathrm{Prop}_U \Omega and $B:S\to {\mathrm{Prop}}_U\Omega$ B:S \to \mathrm{Prop}_U \Omega, if $a \lhd A$ and for all elements $x:S$, $x \lhd B$ and $x{\in }_S^U{\in }_{S}A$ x \in_S^U \in_S A, then $a \lhd B$
   $$\prod _{a:S}\prod _{A:S\to {\mathrm{Prop}}_U\Omega }\prod _{B:S\to {\mathrm{Prop}}_U\Omega }((a⊲A)\times \prod _{x:S}(x{\in }_S^U{\in }_{S}A)\times (x⊲B))\to (a⊲B)$$ \prod_{a:S} \prod_{A:S \to \mathrm{Prop}_U} \Omega} \prod_{B:S \to \mathrm{Prop}_U} \Omega} \left((a \lhd A) \times \prod_{x:S} (x \in_S^U \in_S A) \times (x \lhd B)\right) \to (a \lhd B)
4. For all elements $a:S$ and $b:S$ and subsets $A:S\to {\mathrm{Prop}}_A\Omega$ A:S \to \mathrm{Prop}_A \Omega, if $a \lhd A$ or $b \lhd A$, then $a \cap b \lhd A$.
   $$\prod _{a:S}\prod _{b:S}\prod _{A:S\to {\mathrm{Prop}}_U\Omega }((a⊲A)\times (b⊲A))\to (a\cap b⊲A)$$ \prod_{a:S} \prod_{b:S} \prod_{A:S \to \mathrm{Prop}_U} \Omega} ((a \lhd A) \times (b \lhd A)) \to (a \cap b \lhd A)
5. For all elements $a:S$ and subsets $A:S\to {\mathrm{Prop}}_U\Omega$ A:S \to \mathrm{Prop}_U \Omega and $B:S\to {\mathrm{Prop}}_U\Omega$ B:S \to \mathrm{Prop}_U \Omega, if $a \lhd A$ and $a \lhd B$, then
   $$a⊲\sum _{z:S}\prod _{x:S}\prod _{y:S}(x{\in }_S^U{\in }_{S}A)\times (y{\in }_S^U{\in }_{S}B)\times (z{=}_{S}x\cap u)$$ a \lhd \sum_{z:S} \prod_{x:S} \prod_{y:S} (x \in_S^U \in_S A) \times (y \in_S^U \in_S B) \times (z =_S x \cap u)
   $$\prod _{a:S}\prod _{A:S\to {\mathrm{Prop}}_U\Omega }\prod _{B:S\to {\mathrm{Prop}}_U\Omega }(a⊲A)\times (a⊲B)\to (a⊲\sum _{z:S}\prod _{x:S}\prod _{y:S}(x{\in }_S^U{\in }_{S}A)\times (y{\in }_S^U{\in }_{S}B)\times (z{=}_{S}x\cap y))$$ \prod_{a:S} \prod_{A:S \to \mathrm{Prop}_U} \Omega} \prod_{B:S \to \mathrm{Prop}_U} \Omega} (a \lhd A) \times (a \lhd B) \to \left(a \lhd \sum_{z:S} \prod_{x:S} \prod_{y:S} (x \in_S^U \in_S A) \times (y \in_S^U \in_S B) \times (z =_S x \cap y)\right)
6. For all elements $a:S$, $a \lhd \{\top\}$,
   $$\prod_{a:S} a \lhd \{\top\}$$
7. For all elements $a:S$ and subsets $A:S\to {\mathrm{Prop}}_U\Omega$ A:S \to \mathrm{Prop}_U \Omega, if $\Diamond a$ and $a \lhd A$, there exists an element $x:S$ such that $x \in_S A$ and $\Diamond x$.
   $$\prod _{a:S}\prod _{A:S\to {\mathrm{Prop}}_U\Omega }(\diamond a)\times (a⊲A)\to \sum _{x:S}(x{\in }_{S}A)\times (\diamond x)$$ \prod_{a:S} \prod_{A:S \to \mathrm{Prop}_U} \Omega} (\Diamond a) \times (a \lhd A) \to \sum_{x:S} (x \in_S A) \times (\Diamond x)
8. For all elements $a:S$ and subsets $A:S\to {\mathrm{Prop}}_U\Omega$ A:S \to \mathrm{Prop}_U \Omega, if $\Diamond a$ implies $a \lhd A$, then $a \lhd A$.
   $$\prod _{a:S}\prod _{A:S\to {\mathrm{Prop}}_U\Omega }(\diamond a\to a⊲A)\to (a⊲A)$$ \prod_{a:S} \prod_{A:S \to \mathrm{Prop}_U} \Omega} (\Diamond a \to a \lhd A) \to (a \lhd A)

References

• Mike Fourman and Grayson (1982); Formal Spaces. This is the original development, intended as an application of locale theory to logic.

• Giovanni Sambin (1987); Intuitionistic formal spaces; pdf.

  • This is the probably the main reference on the subject.
  • Warning: you can ignore the material about foundations and type theory, but if you do read any of it, the term ‘category’ means (roughly) proper class; this was common in type theory in the 1980s.

• Giovanni Sambin (2001); Some points in formal topology; pdf. This has newer results, alternative formulations, etc.

• Erik Palmgren, From Intuitionistic to Point-Free Topology: On the Foundation of Homotopy Theory, Logicism, Intuitionism, and Formalism Volume 341 of the series Synthese Library pp 237-253, 2005 (pdf)