Pointless topology

In mathematics, pointless topology (also called point-free or pointfree topology, or locale theory) is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions.^[1]

This revolutionary idea suggests that constructing topologically interesting spaces from purely algebraic data is possible.^[2] The first approaches to topology were geometrical, one started from Euclidean space and patched things together. But Stone's work showed that topology can be viewed from an algebraic point of view (lattice-theoretic). Apart from Stone, Henry Wallman was the first person to exploit this idea. Others continued this path till Charles Ehresmann and his student Jean Bénabou (and simultaneously others), made the next fundamental step in the late fifties. Their insights arose from the study of "topological" and "differentiable" categories.^[2]

Ehresmann's approach involved using a category whose objects were complete lattices which satisfied a distributive law and whose morphisms were maps which preserved finite meets and arbitrary joins. He called such lattices "local lattices", others like Dowker called them "frames" to avoid ambiguity with other notions in lattice theory.^[3]

Intuitively[]

Traditionally, a topological space consists of a set of points together with a topology, a system of subsets called open sets that with the operations of intersection and union forms a lattice with certain properties. Point-free topology is based on the concept of a "realistic spot" instead of a point without extent. Spots can be joined (forming a complete lattice) and if a spot meets a join of others it has to meet some of the constituents, which, roughly speaking, leads to the distributive law

${\displaystyle b\wedge \left(\bigvee a_{i}\right)=\bigvee \left(b\wedge a_{i}\right)}$.

Formally[]

The basic concept is that of a frame, a complete lattice satisfying the distributive law above; frame homomorphisms respect all joins (in particular, the least element of the lattice) and finite meets (in particular, the greatest element of the lattice).

Frames, together with frame homomorphisms, form a category.

Relation to point-set topology[]

In classical topology, represented on a set ${\displaystyle X}$ by the system ${\displaystyle \Omega (X)}$ of open sets, ${\displaystyle \Omega (X)}$ (partially ordered by inclusion) is a frame, and if ${\displaystyle f\colon X\to Y}$ is a continuous map, ${\displaystyle \Omega (f)\colon \Omega (Y)\to \Omega (X)}$ defined by ${\displaystyle \Omega (f)(U)=f^{-1}[U]}$ is a frame homomorphism. For sober spaces such ${\displaystyle \Omega (f)}$ are precisely the frame homomorphisms ${\displaystyle h\colon \Omega (Y)\to \Omega (X)}$. Hence ${\displaystyle \Omega }$ is a full embedding of the category of sober spaces into the dual of the category of frames (usually called of the category of locales). This justifies thinking of frames (locales) as of generalized topological spaces. A frame is spatial if it is isomorphic to a ${\displaystyle \Omega (X)}$. There are plenty of non-spatial ones and this fact turned out to be helpful in several problems.

The theory of frames and locales[]

The theory of frames and locales in the contemporary sense was initiated in the late 1950s (Charles Ehresmann, Jean Bénabou, Hugh Dowker, Dona Papert) and developed through the following decades (John Isbell, Peter Johnstone, Harold Simmons, Bernhard Banaschewski, Aleš Pultr, Till Plewe, Japie Vermeulen, Steve Vickers) into a lively branch of topology, with application in various fields, in particular also in theoretical computer science. For more on the history of locale theory see.^[4]

It is possible to translate most concepts of point-set topology into the context of locales, and prove analogous theorems. Regarding the advantages of the point-free approach let us point out, for example, the fact that some important facts of classical topology depending on choice principles become choice-free (that is, constructive, which is, in particular, appealing for computer science). Thus for instance, products of compact locales are compact constructively, or completions of uniform locales are constructive. This can be useful if one works in a topos that does not have the axiom of choice. Other advantages include the much better behaviour of paracompactness, or the fact that subgroups of localic groups are always closed.

Another point where locale theory and topology diverge strongly is the concepts of subspaces versus sublocales: by Isbell's density theorem, every locale has a smallest dense sublocale. This has absolutely no equivalent in the realm of topological spaces.

See also[]

• Heyting algebra. A frame is a complete Heyting algebra.
• Complete Boolean algebra. Any complete Boolean algebra is a frame (it is a spatial frame if and only if it is atomic).
• Details on the relationship between the category of topological spaces and the category of locales, including the explicit construction of the duality between sober spaces and spatial locales, are to be found in the article on Stone duality.
• Point-free geometry.

Citations[]

Bibliography[]

A general introduction to pointless topology is

This is, in its own words, to be read as the trailer for Johnstone's monograph (which appeared already in 1982 and can still be used for basic reference):

• Johnstone, Peter T. (1982). Stone Spaces. Cambridge University Press, ISBN 978-0-521-33779-3.

There is a recent monograph

• Picado, Jorge, Pultr, Aleš (2012). Frames and locales: Topology without points. Frontiers in Mathematics, vol. 28, Springer, Basel.

where one also finds a more extensive bibliography.

For relations with logic:

• Vickers, Steven (1996). Topology via Logic. Cambridge Tracts in Theoretical Computer Science, Cambridge University Press.

For a more concise account see the respective chapters in:

• Pedicchio, Maria Cristina, Tholen, Walter (Eds.). Categorical Foundations - Special Topics in Order, Topology, Algebra and Sheaf Theory. Encyclopedia of Mathematics and its Applications, Vol. 97, Cambridge University Press, 2003, pp. 49–101.
• Hazewinkel, Michiel (Ed.). Handbook of Algebra. Vol. 3, North-Holland, Amsterdam, 2003, pp. 791–857.
• Grätzer, George, Wehrung, Friedrich (Eds.). Lattice Theory: Special Topics and Applications. Vol. 1, Springer, Basel, 2014, pp. 55–88.