Let $X$ and $Y$ be locally convex spaces, and $\varphi: X\to Y$ a linear continuous mapping. Suppose first that $S$ is a compact set in $X$. Then $\varphi$, being considered as a mapping from $S$ to $\varphi(S)$,
$$
\varphi\Big|\_S:S\to \varphi(S)
$$
is *open* in the sense that for any open set $U$ in $S$ (with respect to the topology induced from $X$) its image $\varphi(U)$ is an open set in $\varphi(S)$ (with respect to the topology induced from $Y$).


This is strange, I was sure that the same must be true if $S$ is not necessarily compact, but just totally bounded (since we can consider the extension of $\varphi$ to the completions of the spaces $X$ and $Y$), but recently I understood unexpectedly that I can't write an accurate proof. Does this mean that there is a counterexample?


So my question:



> 
> Is $\varphi\Big|\_S:S\to \varphi(S)$ an open mapping for each totally bounded set $S$ in $X$?
> 
> 
> 



Are linear continuous mappings open on totally bounded sets?
Bounded sets, e.g., unit balls in normed spaces, are always totally bounded for weak topologies. So you only have to find such a ball with two incompatible weak topologies, which is easy to do—say the ball of a suitable dual space with the weak and the weak $\ast$ topology.


