Is the median a "metric" or a "topological" property? I apologize for the slight abuse of terminology; I hope it will become clear what I mean below.
Consider a random variable $X$. Both the mean and the median can be characterized by an optimality criterion: The mean is that number $\mu$ that minimizes $\mathrm E((X - \mu)^2)$, and the median that number which minimizes $\mathrm E(|X - \mu|)$. In this perspective, the difference between mean and median is the choice of "metric" for evaluating deviations, the square or the absolute value.
On the other hand, the median is that number for which $\mathrm{Pr}(X \leq \mu) = \frac12$ (assuming absolute continuity), i.e. this definition depends only on the ability to order values of $X$ and is independent of how much they differ. A consequence of this is that for every strictly increasing function $f(x)$, $\mathrm{median}(f(X)) = f(\mathrm{median}(X))$, meaning it is "topological" in the sense of invariance under "rubber-like" transformations.
Now I've done the math and I know that starting from the optimality criterion I can arrive at the $\frac12$-quantile, so both describe the same thing. But still I am confused, because my intuition tells me that something that depends on a "metric" cannot lead to a "topological" property.
Can someone resolve this riddle for me?
 A: The flaw in your reasoning is that something that depends on a metric cannot be a topological property.
Take compactness of metric spaces.  This can be defined in terms of the metric: compactness means that the space is complete (depends on the metric) and totally bounded (depends on the metric).  It turns out though, that this property is an invariant under homeomorphism, and indeed, can be defined in terms of only the topology (finite sub covers of any cover, the usual way).
Another example is the various homology theories.  Only singular homology is truly topological in its definition.  All the others, simplicial, cellular, De Rham (cohomology, but grant me a little looseness), etc, depend on extra structure, but turn out to be equivalent (and quite a bit easier to work with). 
This comes up a lot in math, sometimes the easiest way to go about defining something is in terms of some ancillary structure, and then it is demonstrated that the resulting entity does not, in fact, depend on the choice of ancillary structure at all.
