Given a topological space X and an elementary submodel M we
can define a new topological space XM. We investigate for
which topological properties P it is true that if XM has P, then
X has P. We first look at this question in general and then
we impose conditions on M.
In particular, we show some preservation results assuming M to be
\omega-covering and we also show that, under CH, the properties of
being \omega-covering and countably closed are equivalent
for any elementary submodel M. After, we investigate how much
we can weaken the hypothesis of M being \omega-covering.
