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 $.