Let $X,Y$ be topological spaces. We call a continuous map $u:X\to Y$ *universal* if for every continous map $f:X\to Y$ there is $x\in X$ such that $f(x) = u(x)$.


If $u:X\to Y$ and $v:Y\to Z$ are universal, is $v\circ u: X\to Z$ universal?


(I am thinking about this question because of an inspiration from [Fixed points and universal maps for posets](https://mathoverflow.net/questions/180252/fixed-points-and-universal-maps-for-posets) ).



Universal maps between topological spaces
This seems to be answered negatively in "On the composition and products of universal mappings" by W. Holsztyński (Fundamenta Mathematicae 64(2) (1969), 181-188).


I've not looked at the proof or really tried to figure out why it works, but if I understand correctly what is claimed, then one counterexample from the paper is as follows:


$M$ is a Möbius band, regarded as the annulus
$\left\{z\in\mathbb{C}:\frac{1}{2}\leq\vert z\vert\leq 1\right\}$
with $z$ and $-z$ identified for $\vert z\vert=\frac{1}{2}$.


$Q$ is the closed unit disc $\left\{z\in\mathbb{C}:\vert z\vert\leq 1\right\}$.


Apparently the map $u:M\to Q$ induced by $z\mapsto\left(2-\frac{1}{\vert z\vert}\right)z$, which identifies the inner boundary of the annulus to a point, and the map $v:Q\to Q$ given by $z\mapsto z^2$ are both universal, but $v\circ u$ is not.


