Pointless topology

In

History

The first approaches to topology were geometrical, where one started from

Ehresmann's approach involved using a category whose objects were

The theory of

Intuition

Traditionally, a

joined

(symbol ${\displaystyle \vee }$), akin to a union, and we also have a

meet

operation for spots (symbol ${\displaystyle \land }$), akin to an intersection. Using these two operations, the spots form a complete lattice. If a spot meets a join of others it has to meet some of the constituents, which, roughly speaking, leads to the distributive law

${\displaystyle b\wedge \left(\bigvee _{i\in I}a_{i}\right)=\bigvee _{i\in I}\left(b\wedge a_{i}\right)}$

where the ${\displaystyle a_{i}}$ and ${\displaystyle b}$ are spots and the index family ${\displaystyle I}$ can be arbitrarily large. This distributive law is also satisfied by the lattice of open sets of a topological space.

If ${\displaystyle X}$ and ${\displaystyle Y}$ are topological spaces with lattices of open sets denoted by ${\displaystyle \Omega (X)}$ and ${\displaystyle \Omega (Y)}$, respectively, and ${\displaystyle f\colon X\to Y}$ is a

of an open set under a continuous map is open, we obtain a map of lattices in the opposite direction: ${\displaystyle f^{*}\colon \Omega (Y)\to \Omega (X)}$. Such "opposite-direction" lattice maps thus serve as the proper generalization of continuous maps in the point-free setting.

Formal definitions

The basic concept is that of a frame, a

.

The opposite category of the category of frames is known as the category of locales. A locale ${\displaystyle X}$ is thus nothing but a frame; if we consider it as a frame, we will write it as ${\displaystyle O(X)}$. A locale morphism ${\displaystyle X\to Y}$ from the locale ${\displaystyle X}$ to the locale ${\displaystyle Y}$ is given by a frame homomorphism ${\displaystyle O(Y)\to O(X)}$.

Every topological space ${\displaystyle T}$ gives rise to a frame ${\displaystyle \Omega (T)}$ of open sets and thus to a locale. A locale is called spatial if it isomorphic (in the category of locales) to a locale arising from a topological space in this manner.

Examples of locales

• As mentioned above, every topological space ${\displaystyle T}$ gives rise to a frame ${\displaystyle \Omega (T)}$ of open sets and thus to a locale, by definition a spatial one.
• Given a topological space ${\displaystyle T}$, we can also consider the collection of its regular open sets. This is a frame using as join the interior of the closure of the union, and as meet the intersection. We thus obtain another locale associated to ${\displaystyle T}$. This locale will usually not be spatial.
• For each ${\displaystyle n\in \mathbb {N} }$ and each ${\displaystyle a\in \mathbb {R} }$, use a symbol ${\displaystyle U_{n,a}}$ and construct the free frame on these symbols, modulo the relations

${\displaystyle \bigvee _{a\in \mathbb {R} }U_{n,a}=\top \ {\text{ for every }}n\in \mathbb {N} }$

${\displaystyle U_{n,a}\land U_{n,b}=\bot \ {\text{ for every }}n\in \mathbb {N} {\text{ and all }}a,b\in \mathbb {R} {\text{ with }}a\neq b}$

${\displaystyle \bigvee _{n\in \mathbb {N} }U_{n,a}=\top \ {\text{ for every }}a\in \mathbb {R} }$

(where ${\displaystyle \top }$ denotes the greatest element and ${\displaystyle \bot }$ the smallest element of the frame.) The resulting locale is known as the "locale of surjective functions ${\displaystyle \mathbb {N} \to \mathbb {R} }$". The relations are designed to suggest the interpretation of ${\displaystyle U_{n,a}}$ as the set of all those surjective functions ${\displaystyle f:\mathbb {N} \to \mathbb {R} }$ with ${\displaystyle f(n)=a}$. Of course, there are no such surjective functions ${\displaystyle \mathbb {N} \to \mathbb {R} }$, and this is not a spatial locale.

The theory of locales

We have seen that we have a functor ${\displaystyle \Omega }$ from the

of the category of sober spaces and continuous maps into the category of locales. In this sense, locales are generalizations of sober spaces.

It is possible to translate most concepts of

, with arbitrary products of paracompact locales being paracompact, which is not true for paracompact spaces, or the fact that subgroups of localic groups are always closed.

Another point where topology and locale theory diverge strongly is the concepts of subspaces versus sublocales, and density: given any collection of dense sublocales of a locale ${\displaystyle X}$, their intersection is also dense in ${\displaystyle X}$.[6] This leads to Isbell's density theorem: every locale has a smallest dense sublocale. These results have no equivalent in the realm of topological spaces.

See also

• Heyting algebra. Frames turn out to be the same as complete Heyting algebras (even though frame homomorphisms need not be Heyting algebra homomorphisms.)
• 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 equivalence between sober spaces and spatial locales, can be found in the article on Stone duality.
• Whitehead's point-free geometry.
• Mereotopology.

References

Bibliography

A general introduction to pointless topology is

• ISSN 0273-0979
  . Retrieved 2016-05-09.

This is, in its own words, to be read as a trailer for Johnstone's monograph and which can be used for basic reference:

• 1982:
  ISBN 978-0-521-33779-3
  .

There is a recent monograph

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

For relations with logic:

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

For a more concise account see the respective chapters in:

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