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.
CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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.
A formal topology or formal space is a set $S$ together with
such that
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 $a \cap b$ and 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.
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.)
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.
Giovanni Sambin (2001); Some points in formal topology; pdf. This has newer results, alternative formulations, etc.