Topology, in its most abstract sense, is the study of a family of subsets, called open sets, of some given set , when subject to certain conditions based purely on set-theoretic operations. Namely, these conditions are that the intersection of two open sets is an open set, union of open sets is an open set, and that the empty set and are themselves open sets. One can thus think of a topology as a certain kind of a lattice of subsets associated with the set , and the study of a general topological space can be distilled further, to the study of this particular lattice . In this setting, the basic elements under scrutiny are no longer “points” in , but elements of . This shift in focus is the starting point of the so-called “pointless topology”, where the study of topology takes on a lattice-theoretic flavor.
for any , where .
A “pointless” proof. To see that the lattice of open sets of a topological space is a locale, we first observe that is a complete lattice, since arbitrary joins of open sets are open by definition, which is enough to ensure that arbitrary meets exist too (although they are not arbitrary intersections, they are interiors of the intersections). This means that and are both open sets, and hence the expressions in the equality are at least meaningful at this point. Further, , because for individual . So we are left with showing that . Again, this is true at an individual level:
Let’s write . Then the expression above becomes , where are all open sets. Put it another way,
Taking the intersection with on both sides, we have
Substituting and back to their original form, we have , which is what we wanted to prove.
Notice that in the proof above, no points of are employed, and everything is done via basic set operations, as well as extra set operations, such as the interior operator.
Another thing worthy of note is the following fact:
A sketch of proof goes as follows: if is a locale, then for any , the element is the relative pseudocomplement of in . Conversely, if is a Heyting algebra, then we can use the trick that for every , , to show .
Other examples of locales found in topology is the lattice of regular open sets in a topological space .
A locale is also called a frame, and a frame homomorphism a locale homomorphism. However, P. T. Johnstone distinguishes the two names when they are considered as categories. The category Fra of frames consists of locales (or frames) as objects, and locale homomorphisms as morphisms. On the other hand, the category Loc of locales is defined as the opposite category of Fra.
In , a frame and a locale are two distinct objects. A frame is defined as above. But a locale is a triple , where is a frame, is the set of frame homomorphisms from to , the trivial frame (= the trivial Boolean algebra), and is a subset (relation) of given by iff . For each , let . It is not hard to see that the collection of all sets of the form forms a topology on . As a result, every frame can be viewed as a topology on some set!
- 1 F. Borceux, Handbook of Categorical Algebra 3: Categories of Sheaves, Cambridge University Press (1994).
- 2 G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003).
- 3 P. T. Johnstone, The Art of Pointless Thinking: A Student’s Guide to the Category of Locales, Category Theory at Work (Bremen, 1990), pp 85-107, Res. Exp. Math., 18, Heldermann, Berlin (1991).
- 4 P. T. Johnstone, Stone Spaces, Cambridge University Press, Cambridge (1982).
- 5 S. Vickers, Topology via Logic, Cambridge University Press, Cambridge (1989).
|Date of creation||2013-03-22 16:38:11|
|Last modified on||2013-03-22 16:38:11|
|Last modified by||CWoo (3771)|