*Definition 1 (Hamkins).*
Suppose $V \subseteq W$ are transitive models of $\mathrm{ZFC}$ and $\delta$ is a cardinal in $W$.

- $(V,W)$ has the
*$\delta$-cover property*iff for each $A \in W$ with $A \subseteq V$ and $\operatorname{card}^{V}(A) < \delta$ there is some $B \supseteq A$ with $B \in V$ and $\operatorname{card}^{V}(B) < \delta$. $(V,W)$ has the

*$\delta$-approximation property*iff for each $B \in W$ with $B \subseteq V$:If $B \cap C \in V$ for all $C \in V$ with $\operatorname{card}^{V}(C) < \delta$, then $B \in V$.

The standard covering argument shows:

*Lemma 2.*
Let $\delta$ be a regular cardinal, let $\mathbb{P}$ be a forcing
with the $\delta$-c.c. and let $g$ be $\mathbb{P}$-generic. Then
$(V,V[g])$ satisfies the $\delta$-cover property.

With regards to the approximation property, we have:

*Theorem 3 (Hamkins).*
Let $\delta$ be a cardinal, let $\mathbb{P}$ be a nontrivial forcing
of size $\le \delta$ and let $\dot{\mathbb{Q}}$ be such that
$$
\Vdash_{\mathbb{P}} \dot{\mathbb{Q}} \text{ is } \le \delta \text{ strategically closed.}
$$
Then any forcing extension via $\mathbb{P} * \dot{\mathbb{Q}}$
satisfies the $\delta^{+}$ approximation (and the $\delta^{+}$-cover)
property.

*Corollary 4.*
Let $\delta$ be a cardinal, let $\mathbb{P}$ be a forcing of size
$\le \delta$ and let $g$ be $\mathbb{P}$-generic. Then $(V,V[g])$
satisfies the $\delta^{+}$-cover property.

*Question 5.*
Suppose $\delta$ is a cardinal and $\mathbb{P}$ is a $\delta$-closed
forcing. Let $g$ be $\mathbb{P}$-generic. Does $(V,V[g])$ satisfy
the $\delta^{+}$-approximation property?

If the answer is positive: What if $\mathbb{P}$ is only $\kappa$-strategically closed?

In light of Hamkins' theorem the question essentially reduces to:

*Question 6.*
In Hamkins' theorem is it crucial that $\mathbb{P}$ be nontrivial?

Looking at its proof, it seems to me that the answer to this question should be 'no' but Hamkins' phrasing of his theorem makes me wonder whether I missed something. In any case, the main question I'm interested in is:

Question 7.Are there other known properties of a forcing $\mathbb{P}$ to guarantee that its forcing extensions satisfy the $\delta$-approximation property for small $\delta$ compared to $\operatorname{card}(\mathbb{P})$?

This question is intentionally broad. If it turns out that there is an overwhelming wealth of known results, I'd be happy to phrase it more carefully.

Lastly, I'm also interested in *strong negative answers*, should they be known. But I'm not entirely sure how to phrase some such question.

*References.*

Hamkins. Extensions with the approximation and cover properties have no new large cardinals