Let $B$ be a topological space. 
Call a subset $A\subset B$ *ultrafilter-like* iff $A$ is dense in $B$ and
each decomposition $A=A\_1\cup A\_2$, 
into the union of two open subsets extends to a decomposition of
$B=B\_1\cup B\_2$, $A\_1\subset B\_1$ and $A\_2\subset B\_2$ into the union of two open subsets $B\_1$ and $B\_2$.


Is the following true? Let $f:X\rightarrow Y$ be a proper map
and $A\subset B$ is ultrafilter-like.
For any $g:A\rightarrow X$ and $h:B\rightarrow Y$ such that
$f(g(a))=h(a)$ for each $a\in A$, there is a map $g':B\rightarrow X$
extending $g$ such that $f(g'(b))=h(b)$ for each $b\in B$.


Is this true if $Y$ is a point (and thus $X$ is an arbitrary 
quasi-compact space). 


This is true provided $B=A\cup \{\omega\}$
where $\omega$ is closed. If $A$ is discrete, then the neighbourhoods of $\omega$
define an ultrafilter on $A$ and the property becomes
the definition of a proper map via ultrafilters (see Bourbaki, General Topology, I\S10.2, Theorem I).
 If $A$ is not necessarily discrete, then there is an ultrafilter $F$ on $A$
such that the neighbourhoods of $\omega$ are $F$-big,
and the same arguments works. (To fix the terminology: by a proper map
I mean a closed map such that the preimage of any point is quasi-compact).



a characterisation of proper maps via ultrafilters
Let $C$ be a connected Tychonoff space and $a,b\in \beta C\setminus C$ be two distinct points. Let $X=\beta C$, $Y=X/\{a,b\}$ be the quotient space and $f:X\to Y$ be the quotient map. It is clear that $f$ is perfect and hence proper. 


Let $B=Y$, $A=B\setminus \{q(a)\}$, $h:B\to Y$ be the identity map, $g=q^{-1}|A:A\to X\setminus\{a,b\}\subset X$ be the homeomorphism of $A=Y\setminus\{q(a)\}$ onto $X\setminus\{a,b\}$. It is easy to see that $f\circ g=h|A$. On the other hand, it can be shown that $g$ admits no continuous extension to a map $g':B\to X$ such that $f\circ g'=h$. 


The space $g(A)$ is connected sinse it contains a dense connected subspace $C$. Then the space $A$ is connected as well (being homeomorphic to $g(A)$). The connectedness of $A$ implies that $A$ is ultrafilter-like in $B$ (being connected the space $A$ admits no non-trivial partitions into two open sets).


So, to get a sensible answer, we should assume that the space $A$ is disconnected, or better (strongly) zero-dimensional. In this case there is a hope for the positove answer since ultrafilter-likenes of $A$ in $B$ should imply that the identity embedding $A\to\beta A$ extends to an embedding of $B$ into $\beta(A)$.


