For some fixed positive integer $m$, let $\Omega$ be a non-empty open subset of $\mathbb{R}^m$. Let $C(\Omega)$ denote the space of all continuous complex-valued functions on $\Omega$.
For any compact subset $K$ of $\Omega$, let $p_{K}:C(\Omega)\to\mathbb{R}$ be the seminorm defined by $p_{K}(f)=\sup_{x\in K}|f(x)|$. Then it is clear that $\{p_{K}: K\text{ is a compact subset of }\Omega\}$ is a $\textit{separating}$ family of seminorms on $C(\Omega)$. Then there exists a topology $\mathscr{T}$ on $C(\Omega)$ such that $C(\Omega)$ is a locally convex topological vector space with respect to $\mathscr{T}$.
For every $n\in\mathbb{N}$, let $K_n$ be a compact subset of $\Omega$ such that we get an $\textit{Exhaustion}$ of $\Omega$ by compact sets $K_n$, i.e., every $K_n$ is contained in the interior of $K_{n+1}$, $\Omega=\cup_{n\in\mathbb{N}}K_n$ and every compact set in $\Omega$ is contained in some $K_n$. As defined above, we again let $p_n:C(\Omega)\to\mathbb{R}$ to be the seminorm defined by $p_n(f)=\sup_{x\in K_n}|f(x)|$. Then it is again immediate that $\{p_{n}:n\in\mathbb{N}\}$ is a $\textit{separating}$ family of seminorms on $C(\Omega)$, and there exists a topology $\mathcal{T}$ on $C(\Omega)$ such that $C(\Omega)$ is a locally convex topological vector space with respect to $\mathcal{T}$. In fact, in this situation, a translation-invariant, $\mathcal{T}$- compatible metric $d$ can be defined in terms of $\{p_n\}$ by setting $$ d(f,g)= \max_{n\in\mathbb{N}}\frac{2^{-n}p_n(f-g)}{1+p_n(f-g)} \text{ ; where }f,g\in C(\Omega). $$
$\textbf{My question is :}$ can we say that the topologies $\mathscr{T}$ and $\mathcal{T}$ on $C(\Omega)$ are same ?
Intuitively, it seems that the topology $\mathscr{T}$ on $C(\Omega)$ is the same as the topology $\mathcal{T}$ on $C(\Omega)$. Unfortunately, I am not able to think any concrete way to show that $\mathscr{T}=\mathcal{T}$.
I would be grateful for any hint or help. Thanks in advance.
Hint. Note first that each $\mathcal T$-open set is also $\mathscr T$-open (the defining seminorm family of $\mathcal T$ is a subset of that of $\mathscr T$), that is $\mathcal T \subseteq \mathscr T$.
For the other inclusion, note that it suffices to show that each $p_K$ is bounded by some $p_n$ (why?). This amounts to show that each $K$ is contained in some $K_n$. Here use that the $K_n$ are exhausting, write $X = \bigcup_n K_n^o$ and use compactness of $K$.