# Extension of refined subspace topology

Let $$(X,\tau)$$ be a topological space and $$Y$$ be a non-empty subset. Suppose that $$Y$$ is dense in $$(X,\tau)$$ and that there exists a topology $$\tau^{\star}$$ on $$Y$$ which is strictly finer than the subspace topology induced by restriction of $$\tau$$.

Does there exist a topology $$\tau'$$ on $$X$$ whose restriction to $$Y$$ is $$\tau^{\star}$$ but such that $$Y$$ is dense in $$(X,\tau')$$?

## 1 Answer

Let $$\mathcal{U} = \tau \cup \tau^\star$$, and let $$\tau'$$ be the unique minimal topology on $$X$$ containing $$\mathcal{U}$$. Since $$\tau$$ and $$\tau^\star$$ are topologies, they are closed under finite intersection; and since $$\tau^\star$$ is finer than the subspace topology on $$Y$$, the intersection of a set in $$\tau$$ with a set in $$\tau^\star$$ is again in $$\tau^\star$$. Thus $$\mathcal{U}$$ is closed under finite intersection. It follows that $$\tau' = \{ \mbox{all unions of sets in \mathcal{U}}\}.$$ Accordingly, every set $$W$$ in $$\tau'$$ can be written (nonuniquely) in the form $$W = U \cup V$$, where $$U\in \tau$$ and $$V\in \tau^\star$$.

Now if $$x\in X$$ and $$x\in W\in \tau'$$, write $$W = U \cup V$$ as above. If $$x\in U$$, then since $$U\in \tau$$ and $$Y$$ is $$\tau$$-dense in $$X$$, $$U\cap Y \neq \varnothing$$; if $$x\in V$$, then $$V$$ is a nonempty subset of $$Y$$. Taken together we see that $$W \cap Y \neq \varnothing$$, so $$Y$$ is $$\tau'$$-dense in $$X$$.

• I've given this answer some though and does this construction correspond to the initial topology on $X$ corresponding to the identity maps to $\{(X,\tau),(X,\tau^{\star,+}\}$ where $\tau^{\star,+}$ is the minimal topology on $X$ containing $\tau^{\star}\cup \{X\}$ (which if I'm not mistaken is just $\tau^{\star}\cup \{X\}$ (since $Y$ is a subset of $X$ so intersections are just again in $\tau^{\star}$).... and in the special case where $X=Y$ $\tau'$ is the supremum of $\tau^{\star}$ and $\tau$ (which in this trivial case just reduces to $\tau^{\star}$?
– ABIM
Commented Apr 6, 2020 at 18:28