Terminology for dual notion of continuity

Question

Recall the definition of a quotient mapping $f:X\to Y$ between two topological spaces.

It is the following.

1. $f$ is surjective.
2. $f$ is continuous.
3. For each subset $B$ of $Y$, if $f^{-1}(B) $ is open, then $B$ is open.

Question: Have you seen any attempt to introduce terminology for condition (3)? If not, what terminology would you introduce? What immediately comes to my mind are the following: co-continous, dual-continous, reverse-continuous, quotient, saturated-mapping. However none are too enticing over the other and some conflict with standard terminology (co-continous and quotient).

Related Question: Have you seen terminology for the following similar notion dual to being an open map?

• For each subset $A$ of $X$, if $f(A)$ is open, then $A$ is open.

In summary, I am interested in the "dual" notions of continuity and open-mapping. After some online research, I could not find the desired terminologies. I think if I had a better understanding of mathematical etymology, then maybe one of my suggestions would be more natural/enticing. Hence the questions.

Answer 1

Note that a 1-1 continuous function has your property: if $f[A]$ is open, and $f$ is continuous, then $A = f^{-1}[f[A]]$ (which holds by being injective) is open by continuity.

Also, $(X,\mathcal{T}_X) \to (Y, \mathcal{T}_Y)$ has the property iff $X$ has the finest topology that makes $f$ an open map. This is analogous to (in this context) $f$ being quotient means that $Y$ has the finest topology making $f$ continuous. Note that the coarsest (minimal) topology that makes a function open, is the indiscrete one (for any $f$), so that notion is not interesting.

So maybe a good name for this property would be "maximally open function"? I haven't seen it defined before, as such, but the notion has been thought of:

In a book by R. Vaidyanathaswamy, called set topology, on page 119 he considers different topologies on a set $Y$, for a given surjective function $f:(X,\mathcal{T}_X) \to Y$:

• the finest topology $\tau_1$ on $Y$ that makes $f$ continuous. This is now usually called the final topology w.r.t. $f$, but he defines a notion of "strong continuity" and then its the unique topology on $X$ that makes $f$ strongly continuous. We'd call that a quotient map nowadays. (The book is from 1947, so terminology wasn't as standardized. e.g. he calls a topology stronger if it has fewer open sets, so that's confusing to a modern reader.)

• The coarsest topology $\tau_2$ on $Y$ that makes $f$ open. (Generated by the subbase $\{f[O], O \in \mathcal{T}\}$.

• The coarsest topology $\tau_3$ on $Y$ that makes $f$ closed (similarly generated by a closed subbase).

He notes that the latter two are both in general finer than the first. E.g. if $O \in \tau_1$, $f^{-1}[O]$ is open in $X$ but then $O = f[f^{-1}[O]]$ (by surjectivity) is open in $\tau_2$. So $\tau_1 \subseteq \tau_2$. Changing open in closed sets in the last argument gives $\tau_1 \subseteq \tau_3$.

In section 40 he then studies characterisations of the quotient space, where we have a set of classes under an equivalence relation, and we have these three "natural" choices of a topology on the quotient space. Nowadays we use topology $\tau_1$, the quotient topology only. But making $q$ maximally open is also an idea. It's the only place that I know of where such a notion is considered. There is no name for the notion ($f[A]$ open implies $A$ open) itself.