Quotient map, quotient topology in Banach spaces

Question

In Lindenstrauss and Tzafriri's Classical Banach Spaces I an operator $T:X\to Y$ is called a quotient map if the $\overline{TB_X}=B_Y$ where $B_X$ and $B_Y$ are the unit balls in Banach spaces $X$ and $Y$ respectively. The overline denotes closure.

I'm wondering how this definition relates to quotient map and quotient topology in the more abstract setting of topological spaces. For instance, if that condition is met, does $Y$ have the quotient topology induced by $T$? And conversely, if $T:X\to Y$ is such that $Y$ has the quotient topology, is that condition (or a similar one) true?

Answer 1

This definition of quotient has metric in mind, since we ask a unit ball to be mapped to unit ball. As such, it has a natural generalization to metric spaces $X,Y$, but not necessarily to general topological spaces. (What's a generalization of "isometry" to topological spaces?)

Let $B(x,r)$ be the open ball of center $x$ and radius $r$, and $\overline{B}(x,r)$ the closed ball of same center and radius. Here are a few slightly different generalizations: we ask that $f:X\to Y$ is such that for all $x\in X$ and all $r>0$

1. $\overline{f(\overline{B}(x,r))}=\overline{B}(f(x),r)$ (this most closely matches the Banach space condition)
2. $f(B(x,r))=B(f(x),r)$ (such $f$ is called a weak submetry)
3. $f(\overline{B}(x,r))=\overline{B}(f(x),r)$ (such $f$ is called a submetry).

The third condition is the strongest, and most pleasant to work with in the geometry of metric spaces. On the other hand, 2nd is easier to establish when we don't have compactness. All of them are strong enough for $Y$ to be isometric to the quotient metric space $X/f$ (notation means we identify points with the same image under $f$). This is not hard to prove.

The converse is false. If $Y$ is a quotient of $X$ by some equivalence relation $\sim$, then the quotient map $f:X\to Y$ may fail 1-3 rather badly. Here is an example: let $X$ be the segment $[0,4]$ with the Euclidean metric. Take its quotient by this equivalence relation: $x\sim x'$ iff $x=x'$ or $x,x'\in [1,3]$. The quotient map sends the closed ball $B(2,1)$ to a point, not to any kind of ball of radius $1$.

Special case when the converse is true: $Y$ is the quotient of $X$ by a group of isometries such that the orbits are closed. In this case, the quotient map is a weak submetry (e.g., page 851 of Handbook of Geometric Topology by R.B. Sher and R.J. Daverman).

Some names to look up in connection to submetries: Berestovskii, Guijarro, Sharafutdinov, Perelman.