For a topology $\mathcal{T}$ on a set $S$, where $\mathcal{T}$ does not have a finite base, I define the *grasp* $g(\mathcal{T})$ to be the least infinite cardinal $\kappa$ such that $\mathcal{T}$ has a base $\mathcal{B}$ with $|\mathcal{B}|= w(\mathcal{T})$ , ($w$ is the usual *weight* function), satisfying $$\mathcal{T}=\{\bigcup V : V\in [\mathcal{B}]^{\leq \kappa}\}.$$

That is, every open set is the union of at most $\kappa$ members of $\mathcal{B}$, and $|\mathcal{B}|$ is minimum among all bases.

The following can be shown by elementary means:

- $g(\mathcal{T})\leq w(\mathcal{T})$. (Obvious).
- $|\mathcal{T}|\leq w(\mathcal{T})^{g(\mathcal{T})}$. (Obvious).
- $g(\mathcal{T})\leq g(D(w(\mathcal{T})))$ where $D(\lambda)$ is discrete space of cardinality $\lambda$.
- If $Y$ is a subspace of $X$ and $w(Y)=w(X)$ then $g(Y)\leq g(X)$.
When $\kappa$ is an infinite cardinal with the discrete or the order topology then

- $\operatorname{cf}(\kappa)\leq g(\kappa)\leq \kappa$.
- If $\kappa$ is a singular strong limit then $g(\kappa)=\operatorname{cf}(\kappa)$.

(This last one helps to distinguish $g$ from other topological cardinal functions.)

I have two questions.

Question 1:Referring to (2) above, $g(T)$ is not necessarily the least cardinal $\lambda$ such that $|\mathcal{T}|\leq w(\mathcal{T})^\lambda$ because if we assume $2^{\omega}=2^{\omega_1}$ and let $\mathcal{T}$ be the discrete topology on $\omega_1$ then $|\mathcal{T}|=\omega_1^{\omega}$ but by (5)(i) $g(\mathcal{T})=\omega_1$. So is there an example like this in $\mathsf{ZFC}$?

Question 2:By (5)(i) we have $\operatorname{cf}(2^{\omega})\leq g(D(2^{\omega}))\leq 2^{\omega}$. What values for $g(D(2^{\omega}))$ other than $2^{\omega}$ are consistent?