Suppose that we have two topological spaces $(X,T_1)$, $(X,T_2)$ such that $T_1\subset T_2$ and $K\subset X$ is compact in $T_1$ topology. Is it also compact in $T_2$ topology ???
I think that the answer is yes, because we can cover $K$ with a finite collection of open sets that belong in $T_1$ topology, but we know that these open sets belong also in $T_2$, so it is going to be compact in $T_2$.
Is this true ??