A subcategory $\mathcal{C}$ of the category $Top$ of topological spaces is a [reflective subcategory](http://en.wikipedia.org/wiki/Reflective_subcategory) if the inclusion functor $i:\mathcal{C}\hookrightarrow Top$ has a left adjoint $R:Top\rightarrow \mathcal{C}$. In other words, for every space $X$, there is a space $RX\in \mathcal{C}$ and a map $r:X\rightarrow RX$ such that for each map $f:X \rightarrow Y$ with $Y\in \mathcal{C}$, there is a unique map $\tilde{f}:RX\rightarrow Y$ such that $\tilde{f}\circ r=f$.


If the reflection map $r$ is always a quotient map, $\mathcal{C}$ is said to be a quotient-reflective subcategory. For example, it seems that the subcategories of $T\_0$ spaces and $T\_2$ spaces are quotient-reflective but the subcategories of completely regular spaces and compact Hausdorff spaces are reflective but not quotient-reflective. I'd like to hear of other examples of quotient-reflective subcategories as well.


There seem to be conditions fully characterizing when a subcategory is reflective. For example, some necessary conditions are mentioned in the answer to this MO [question](https://mathoverflow.net/questions/9504/why-is-top-4-a-reflective-subcategory-of-top-3). Are there also conditions characterizing subcategories which are quotient-reflective?



How do you know when a reflective subcategory of Top is quotient-reflective?
The following theorem (16.8 in Abstract and Concrete Categories, Adamek-Herrlich-Strecker) could be of use.


Let E and M be subclasses of epis and monoes respectively, closed under composition with isomorphisms.
If A is a full subcategory of an (E,M)-factorisable category B, then the following conditions are
equivalent:


(1) A is E-reﬂective in B.


(2) A is closed under the formation of M-sources in B.


In the case that B has products and is E-co-wellpowered, the above conditions are equivalent to:


(3) A is closed under the formation of products and M-subobjects in B.


In the case of TOP, E would be extremal epis, and M monoes. In topological constructs extremal epis are exactly quotients (final and surjective). TOP is extremal epi-mono factorisable, has products and is extremally co-wellpowered. So a full subcategory is quotient reflective in TOP if and only if it is closed under products and (not necessarily initial) subobjects.


The subcategories of completely regular spaces is for example not closed for "adding more opens", so it is, as you said, not quotient reflective.


