# Extension of the refined topology of the subspace

Leave $$mathcal {U} = tau cup tau ^ star$$, and let $$tau & # 39;$$ be the only minimal topology in $$X$$ containing $$mathcal {U}$$. As $$tau$$ Y $$tau ^ star$$ they are topologies, they are closed under finite intersection; and from $$tau ^ star$$ is finer than the subspace topology in $$Y$$, the intersection of a set in $$tau$$ with a set in $$tau ^ star$$ is back in $$tau ^ star$$. So $$mathcal {U}$$ It is closed under a finite intersection. It turns out that
$$tau & # 39; = { mbox {all set unions in mathcal {U} } }.$$
Consequently, each set $$W$$ in $$tau & # 39;$$ it can be written (not uniquely) in the form $$W = U cup V$$,
where $$U in tau$$ Y $$V in tau ^ star$$.

Yes now $$x in X$$ Y $$x in W in tau & # 39;$$, to write $$W = U cup V$$ as previously. Yes $$x in U$$, then from $$U in tau$$ Y $$Y$$ it is $$tau$$-dense in $$X$$, $$U cap Y neq varnothing$$; Yes $$x in V$$, then $$V$$ it is a non-empty subset of $$Y$$. Taken together we see that $$W cap Y neq varnothing$$, so $$Y$$ it is $$tau & # 39;$$-dense in $$X$$.