formal topology

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.



topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Basic homotopy theory

Formal topology


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 SS together with

such that

  1. a=ba = b whenever a{b}a \lhd \{b\} and b{a}b \lhd \{a\},
  2. aUa \lhd U whenever aUa \in U,
  3. aVa \lhd V whenever aUa \lhd U and xVx \lhd V for all xUx \in U,
  4. abUa \cap b \lhd U whenever aUa \lhd U or bUb \lhd U,
  5. a{xy|xU,yV}a \lhd \{ x \cap y \;|\; x \in U,\; y \in V \} whenever aUa \lhd U and aVa \lhd V,
  6. a{}a \lhd \{\top\},
  7. x\Diamond x for some xUx \in U whenever a\Diamond a and aUa \lhd U, and
  8. aUa \lhd U whenever aUa \lhd U if a\Diamond a,

for all aa, bb, UU, and VV.

We interpret the elements of SS as basic opens in the formal space. We call \top the entire space and aba \cap b the intersection of aa and bb. We say that aa is covered by UU or that UU is a cover of aa if aUa \lhd U. We say that aa is positive or inhabited if a\Diamond a. (For a topological space equipped with a strict topological base SS, 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 SS 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 SS to be a preset originally, then use (1) as a definition of equality.
  • We can prove that (S,,)(S,\cap,\top) is a bounded semilattice; if (as the notation suggests) we interpret this as a meet-semilattice, then aba \leq b if and only if a{b}a \lhd \{b\}. Conversely, we could require that (S,,)(S,\cap,\top) be a semilattice originally, then let (1) say that aba \leq b whenever a{b}a \lhd \{b\}.
  • We can prove that a\Diamond a holds iff every cover of aa is inhabited and that a\Diamond a fails iff aa \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 XX be a topological space, and let SS be the collection of open subsets of XX. Let \top be XX itself, and let aba \cap b be the literal intersection of aa and bb for a,bSa, b \in S. Let aUa \lhd U if and only UU is literally an open cover of aa, and let a\Diamond a if and only if aa is literally inhabited. Then (S,,,,)(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 SS be a base for the topology of XX which is strict in the sense that it is closed under finitary intersection. Let the other definitions be as before. Then (S,,,,)(S,\top,\cap,\lhd,\Diamond) is a formal topology.

More generally, let BB be a subbase for the topology of XX, and let SS be the free monoid on BB, that is the set of finite lists of elements of BB (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 aba \cap b be the concatenation of aa and bb, let aUa \lhd U if the intersection of aa is contained in the union of the intersections of the elements of UU, and let a\Diamond a if the intersection of aa is inhabited. Then (S,,,,)(S,\top,\cap,\lhd,\Diamond) is a formal topology.

Let XX be an accessible locale generated by a posite whose underlying poset SS is a (meet)-semilattice. Let \top and \cap be as in the semilattice structure on SS, and let aUa \lhd U if UU contains a basic cover (in the posite structure on SS) of aa. 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.

    • 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)

Revised on April 23, 2017 12:42:53 by tphyahoo? (