For a topology T$\mathcal{T}$ on a set S$S$,when where $T$ doe$\mathcal{T}$ does not have aa finite base, I define the grasp $g(T)$grasp $g(\mathcal{T})$ to be the least infinite cardinal $ k$ such$\kappa$ such that $T$$\mathcal{T}$ has a base $B$$\mathcal{B}$ with $|B|= w(T)$$|\mathcal{B}|= w(\mathcal{T})$ , ($w=$ weight$w$ is the usual weight function), satisfying $T=\{\cup V : V\in [B]^{\leq k}\}$.$$\mathcal{T}=\{\bigcup V : V\in [\mathcal{B}]^{\leq \kappa}\}.$$
That is,every every open set is the union of at most $k$$\kappa$ members of $B$$\mathcal{B}$,and and $|B|$$|\mathcal{B}|$ is minimum among all bases.The
The following can be shown by elementary means : $ (1)g(T)\leq w(T) .$ (Obvious).(2). $|T|\leq w(T)^{g(T)} .$ (Obvious). $ (3).g(T)\leq g(D(w(T))) $ where $D(x) $ is discrete space of cardinality $x . $ (4) If $Y$ is a subspace of $X$ and $w(Y)=w(X)$ then $g(Y)\leq g(X) .$ (5). When $k$ is an infinite cardinal with the discrete or the order topology then (i) $ cf(k)\leq g(k)\leq k .$ (ii) If $ k$ is a singular strong limit then $g(k)=cf(k) . $This last one helps to distinguish $g$ from other topological cardinal functions.I
$g(\mathcal{T})\leq w(\mathcal{T})$. (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 2two questions. Q1: Referring to (2) above, $g(T)$ is not necessarily the least cardinal $l$ such that $|T|\leq w(T)^l$ because if we assume $2^{\omega}=2^{\omega_1}$ and let $T$ be the discrete topology on $\omega_1$ then $|T|=\omega_1^{\omega}$ but by (5)(i) $g(T)=\omega_1$. So is there an example like this in ZFC ? And Q2 : By (5)(i) we have $cf(2^{\omega})\leq g(D(2^{\omega}))\leq 2^{\omega}$. What values for $g(D(2^{\omega}))$ other than $2^{\omega}$ are consistent?
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?
