The name for the quotient property.

Question

We call a surjective $f:X\rightarrow Y$ a quotient mapping if it satisfies, for every $U\subset Y$

(continuity, continuous) $U$ is open $\Rightarrow$ $f^{-1}(U)$ is open

and

(???) $f^{-1}(U)$ is open $\Rightarrow$ $U$ is open.

I was wondering if there is the name for the second property itself of the definition, or should I just call it "the other" property? And if a function satisfies this property only, how should I call it?

Answer 1

Normally we only consider the continuity as given, and then the property that ($f^{-1}[U]$ open implies $U$ is open) is what makes $f$ a quotient map (by definition!).

So a continuous onto $f: X \rightarrow Y$ is defined to be a quotient map, iff for all $U \subset Y$: $f^{-1}[U]$ open in $X$ implies $U$ open.

Or equivalently $f: X \rightarrow Y$ (onto function) is a quotient map iff for all $U \subset Y$: $U$ open iff $f^{-1}[U]$ open.

So your "other" is just the typical property for being a quotient map. Normally this property is not considered without continuity of $f$ as well.

Answer 2

A more general name for this kind of topology is "final." This involves a change of perspective: we imagine the topology on $Y$ is not already given, but that it's going to be defined by the quotient map $q$. We say that a quotient space has the final topology induced by $q$. "Final" here means that the topology is the finest possible one that makes $q$ continuous, that is, we put in as many open sets as possible. What's a possible open set? Well, if $q$ is continuous and $U$ is going to be open, then we know we must have $q^{-1}(U)$ open. It's quick to check that if all such $U$ are declared to be open, $Y$ becomes a topological space admitting a continuous surjection $q$ from $X$, which justifies calling this the final topology.

We can define final topologies induced by more than one map, which is where we begin to generalize. For instance, any object defined by gluing charts together, in particular, vector bundles and manifolds, have the final topology induced by all their charts. This is generally an uncountable collection, so it's a very different situation than a quotient map. For another example, any open map induces the final topology on its image, so that a surjective open map is automatically a quotient map; similarly for a surjective closed map (which is already open.) But this implication is not reversible-there are quotient maps that are neither open nor closed!