CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
A point of a topological space is called focal (Freyd-Scedrov) if its only open neighbourhood is the entire space.
The closed point $\bot$ of Sierpinski space $\mathbf{2}$ is a focal point.
The vertex $v$ of a Sierpinski cone (or scone) $s(X)$ on a space $X$, given by a pushout in $Top$
is a focal point. This construction is in fact the same as generically adding a focal point to $X$.
The prime spectrum of a ring $A$ has a focal point iff $A$ is a local ring. In this case, the focal point is given by the unique maximal ideal of $A$.
The category of sheaves over (the site of open subsets) of a topological space with focal point is a local topos.
Every topos has a free “completion” to a “focal space”, given by its Freyd cover.
A locale $X$ is called local if in any covering of $X$ by opens $U_i$, at least one $U_i$ is $X$.
The locale associated to an sober space is local in this sense if and only if the space possesses a focal point (see Johnstone, discussion preceding lemma C1.5.6). Locale theoretically, this point is then given by the frame homomorphism
where $\Omega$ is the frame of opens of the point.