When is a set open in the topology of pointwise convergence?

Question

If $E$ is a set, then the topology $\rho(E)$ generated by $$p_x(f):=|f(x)|\;\;\;\text{for }f:E\to\mathbb R$$ for $x\in E$ is called the topology of pointwise convergence on $\mathbb R^E$.

Are we able to show that if $\Gamma\subseteq C(E,\tau)$ is $\tau$-equicontinuous, then$^1$ $\left.\rho(E)\right|_\Gamma=\left.\kappa(E,\tau)\right|_\Gamma$? If not, what do we need to assume to show that?

Moreover, I would like to know how exactly we can show that $$\kappa(E,\tau)\subseteq\left.\rho(E)\right|_{C(E,\:\tau)}\tag1.$$

Is there a useful characterization of $\rho(E)$-openness? For example, if $\tau$ is a topology on $E$, is $C(E,\tau)\in\rho(E)$? And given an arbitrary subset $\Gamma\subseteq C(E,\tau)$, how can we show that $\Gamma\in\rho(E)$?

I know that generally, if $X$ is a vector space and $\sigma$ is the topology generated by a family $P$ of seminorms on $X$, then $$\mathcal B_P:=\left\{\varepsilon\bigcap_{p\in F}U_p:F\subseteq P\text{ is finite and }\varepsilon>0\right\}$$ is an analytic basis for $\sigma$, where $$U_p:=\{x\in X:p(x)<1\}\;\;\;\text{for }p\in P.$$

This clearly gives an abstract characterization of being $\rho(E)$-open.

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$^1$ If $(X,\sigma)$ is a topological space and $B\subseteq X$, then $\left.\sigma\right|_B:=\{O\cap B:O\in\sigma\}$ denotes the subspace topology on $B$.

Answer 1

In quite concrete terms $O \subseteq F \subseteq \Bbb R^E$ is open in the topology of pointwise convergence (relative to $F$) iff

For every $f \in O$ there exist finitely many $x_1,\ldots, x_n \in E$ and $r>0$ such that $$\{g \in F\mid \forall 1 \le i \le n: |f(x_i) - g(x_i)| < r \}\subseteq O$$

This is quite standard. Note that it’s just the product-topology on $\Bbb R^E$ and its subsets in mild disguise.