# locale

Topology  , in its most abstract sense, is the study of a family of subsets, called open sets, of some given set $X$, 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 $X$ are themselves open sets. One can thus think of a topology as a certain kind of a lattice   $L(X)$ of subsets associated with the set $X$, and the study of a general topological space can be distilled further, to the study of this particular lattice $L(X)$. In this setting, the basic elements under scrutiny are no longer “points” in $X$, but elements of $L(X)$. 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.

Specifically, the lattice $L(X)$ is an example of what is known as a locale. Formally, a locale is a complete lattice  $L$ that is meet infinitely distributive:

 $x\wedge(\bigvee Y)=\bigvee(x\wedge Y)$

for any $Y\subseteq L$, where $x\wedge Y:=\{x\wedge y\mid y\in Y\}$.

A “pointless” proof. To see that the lattice $L(X)$ of open sets of a topological space $X$ is a locale, we first observe that $L(X)$ 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 $\bigvee Y$ and $\bigvee(x\bigwedge Y)$ are both open sets, and hence the expressions in the equality are at least meaningful at this point. Further, $\bigvee(x\wedge Y)=\bigcup(x\cap Y)\subseteq x\cap(\bigcup Y)=x\wedge(\bigvee Y)$, because $x\cap y\subseteq x\cap(\bigcup Y)$ for individual $y\in Y$. So we are left with showing that $x\wedge(\bigvee Y)\subseteq\bigvee(x\wedge Y)$. Again, this is true at an individual level:

 $\displaystyle x\cap y\subseteq\bigcup(x\cap Y).$ (1)

Let’s write $z=\bigcup(x\cap Y)$. Then the expression above becomes $y\cap x\subseteq z$, where $x,y,z$ are all open sets. Put it another way,

 $\displaystyle y$ $\displaystyle=$ $\displaystyle(y\cap x)\cup(y-x)$ (2) $\displaystyle\subseteq$ $\displaystyle z\cup(y-x)$ (3) $\displaystyle=$ $\displaystyle z\cup(y\cap x^{c})$ (4) $\displaystyle=$ $\displaystyle(z\cup y)\cap(z\cup x^{c})$ (5) $\displaystyle\subseteq$ $\displaystyle(z\cup y)\cap(z\cup x^{c})^{\circ},$ (6)

where $-$ denotes the set difference  operator, ${}^{c}$ is the set complementation, and ${}^{\circ}$ the interior operator. (6) comes from the fact that $y$ is open. Now, take the union of all $y\in Y$, and write this union $t:=\bigcup Y=\bigcup\{y\mid y\in Y\}$. Then

 $\displaystyle t$ $\displaystyle\subseteq$ $\displaystyle(z\cup t)\cap(z\cup x^{c})^{\circ}$ (7) $\displaystyle\subseteq$ $\displaystyle(z\cup t)\cap(z\cup x^{c})$ (8) $\displaystyle=$ $\displaystyle z\cup(t\cap x^{c}).$ (9)

Taking the intersection with $x$ on both sides, we have

 $\displaystyle t\cap x$ $\displaystyle\subseteq$ $\displaystyle(z\cup(t\cap x^{c}))\cap x$ (10) $\displaystyle=$ $\displaystyle(z\cap x)\cup((t-x)\cap x)$ (11) $\displaystyle=$ $\displaystyle(z\cap x)\cup\varnothing$ (12) $\displaystyle=$ $\displaystyle z\cap x$ (13) $\displaystyle\subseteq$ $\displaystyle z.$ (14)

Substituting $t$ and $z$ back to their original form, we have $x\cap(\bigcup Y)\subseteq\bigcup(x\cap Y)$, which is what we wanted to prove.

Notice that in the proof above, no points of $X$ are employed, and everything is done via basic set operations, as well as extra set operations, such as the interior operator.

Remarks.

## References

• 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, , Cambridge University Press, Cambridge (2003).
• 3 P. T. Johnstone, , 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).
Title locale Locale 2013-03-22 16:38:11 2013-03-22 16:38:11 CWoo (3771) CWoo (3771) 12 CWoo (3771) Definition msc 06D22 frame frame homomorphism CompleteHeytingAlgebra locale homomorphism