Contents

topos theory

# Contents

## Idea

The concept of weakly open subtopos is weakening of the concept of a dense subtopos in topos theory.

## Definition

Let $\mathcal{E}$ be a topos and $Sh_j(\mathcal{E})$ a subtopos of $\mathcal{E}$ with $j$ the corresponding Lawvere-Tierney topology, $a_j$ the corresponding associated sheaf functor and $\neg$ and $\neg^j$ the pseudo-complementation operators on the subobject lattices in $\mathcal{E}$ and $Sh_j(\mathcal{E})$, respectively. The subtopos $Sh_j(\mathcal{E})$ is called weakly open if for all subobjects $A\rightarrowtail E$ in $\mathcal{E}$, $a_j(\neg A)\cong \neg^j a_j(A)$.

## Remark

In other words, the sheafification $a_j$ commutes with pseudo-complementation.

If $i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ is weakly open then the topology $j$ and the inclusion $i$ are called weakly open as well.

Since the associated sheaf functor $a_j$ is just the inverse image of the inclusion $i: Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ the definition can be generalized to general geometric morphisms.

## Properties

As already the name suggests, the concept of a weakly open subtopos can equally be viewed as a weakening of the concept of an open subtopos. Indeed, since the associated sheaf functor of an open inclusion is logical we have

###### Proposition

An open subtopos $i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ is weakly open. $\qed$

Less straightforward is the following

###### Proposition

A dense subtopos $i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ is weakly open.

The concept of map in point-set topology that corresponds to weakly open is that of a continuous map $f$ such that $f(U)$ is dense in some open set, for $U$ open (cf. Pták (1958)).