Given an equivalence relation $R$ on a topological space $X$, there are certain conditions we may ask of $R$ that imply certain well-behavedness conditions on the quotient space $X/\mathord{\sim}_R$. One of these conditions which turns out to be of particular use is for $R$ to be an open relation, where we ask that for all $U\in\mathrm{Open}(X)$, we have $R_*(U)\in\mathrm{Open}(X)$ as well, where $$R_*(U)=\{x\in X\ |\ \text{there exists $u\in U$ with $x\sim_Ru$}\}.$$ In this case, we have the following three results, the latter two of which are particularly useful when working with topological manifolds:
- (Clementino–Tholen). The following conditions are equivalent:
- (a) The relation $R$ is open.
- (b) The quotient map $\mathrm{pr}\colon X\to X/\mathord{\sim}_R$ is open.
- (c) The inclusion map $\iota_{X/\mathord{\sim}_R}\colon X/\mathord{\sim}_R\to\mathcal{P}^{-}(X)$ is continuous, where $\mathcal{P}^{-}(X)$ denotes the powerset of $X$ equipped with the lower Vietoris topology.
- (Boothby, Lemma 2.3 of Chapter 3). If $R$ is open and $X$ is second-countable, then so is $X/\mathord{\sim}_{R}$.
- (Boothby, Lemma 2.4 of Chapter 3). If $R$ is open, then $X/\mathord{\sim}_{R}$ is Hausdorff iff $R\subset X\times Y$ is closed in the product topology.
Now, there are three other conditions of a similar flavor we may ask of $R$ (see here for more background, including the general case, some equivalent conditions, and a couple pointers to the literature):
- We can ask that $R$ is closed in that $R_*$ sends closed sets to closed sets.
- We can ask that $R$ is lower semicontinuous in that the function $$R^{-1}\colon\mathcal{P}(X)\to\mathcal{P}(X)$$ given by $$R^{-1}(V)=\{x\in X\ |\ R(x)\cap V\neq\emptyset\}$$ sends open sets to open sets.
- We can ask that $R$ is upper semicontinuous in that $R^{-1}$ sends closed sets to closed sets.
Question. For closed relations, we have an analogous result to Item (1) above (that the projection map is closed, etc.), so both closedness and openness are useful when working with quotients, but what about upper/lower semicontinuity? Are there any results relating upper/lower semicontinuity of $R$ to well-behavedness conditions of the quotient space $X/\mathord{\sim}_R$ or the projection $\mathrm{pr}\colon X\to X/\mathord{\sim}_R$?
