The concept of well-pointed topoi was introduced by Freyd in [1]. A topos is well-pointed if it satisfies either the following equivalent conditions:

1. The terminal object $1$ distinguishes morphisms in the sense that if the diagram $$\xymatrix{ 1\ar[r]^x & A\ar@<1ex>[r]^f\ar@<-1ex>[r]_g & B }$$ commutes for every morphism $x\colon 1\to A$ then in fact $$\xymatrix{ A\ar@<1ex>[r]^f\ar@<-1ex>[r]_g & B }$$ commutes, that is, $f=g$ Moreover, $1$ is not isomorphic to the initial object.
2. The topos $\mathcal{T}$ is complemented and supports split, and the truth object $\Omega$ of $\mathcal{T}$ has exactly two elements, $\top\colon 1\to\Omega$ and $\bot\colon 1\to\Omega$ To say that $\mathcal{T}$ is complemented means that if $m\colon X\to Y$ is a monomorphism, then there exists a monomorphism $m'\colon X'\to Y$ such that $m\sqcup m'\colon X\sqcup X'\to Y$ is an isomorphism. To say that $\mathcal{T}$ supports split means that every subobject of $1$ is projective.

Every well-pointed topos is a Boolean topos.

Bibliography

1

P. Freyd. Aspects of topoi. Bull. Austral. Math. Soc. 7 (1972), 1-76.

2

P. T. Johnstone. Topos theory. Academic Press, 1977.

3

S. Mac Lane and I. Moerdijk. Sheaves and Geometry in Logic: A First Introduction to Topos Theory, Springer-Verlag, 1992.