Showing that the balls given by seminorms form a local base

Question

Let $X$ be Hausdorff and let $\mathcal{P}$ be a (countable) family of seminorms on $X$ which generate a topology on $X$ via the neighbourhoods $B_{p}(x,r):=\{x\in X:p(x)<r\}$, for $p\in\mathcal{P}$ and $r>0$. Now, define a new family of seminorms $\widetilde{\mathcal{P}}$ on $X$ of the form $\widetilde{p}:=\max_{1\le j\le k}p_j$ for $k\in\mathbb{N}$ and $p_j\in\mathcal{P}$.

I want to show that the balls $B_{\widetilde{p}}(x,r_{\widetilde{p}})$ form a local base in $X$. However, I am struggling to prove this, even from an intuitive perspective.

Essentially, we want to show that if $U$ is any neighbourhood of $0$ in $X$, then there exists a $\widetilde{p}\in\widetilde{\mathcal{P}}$ such that $B_{\widetilde{p}}(0,r)\subset U$.

However, due to the construction of our topology, $U$ is of the form $B_p(0,r_p)$ for $p\in\mathcal{P}$ and $r_p>0$. Hence we want to show, in fact, that for all $p_j\in\mathcal{P}$ and $r_{j}>0$, there exists a $\widetilde{p}\in\widetilde{\mathcal{P}}$ and $r_{\widetilde{p}}>0$ such that $B_{\widetilde{p}}(0,r_{\widetilde{p}})\subset B_{p_j}(0,r_j)$ for all $j\in[1,k]$.

But this seems counter intuitive, since for all $p_1,...,p_k\in\mathcal{P}$ and $r_1,...,r_k>0$, there exists $\widetilde{p}\in\widetilde{\mathcal{P}}$ such that $\max\{p_1(x),...,p_k(x)\}=\widetilde{p}(x)<r_{\widetilde{p}}$, which implies that $r_{\widetilde{p}}>r_{j}$ for all $p_j\in\mathcal{P}$ (with $0\le j\le k)$ and therefore the "containment" would be the reverse of what I want to actually prove.

Answer 1

The intuition is that inequalities get reversed at key moments.

Write $\tilde p_k = \max_{1\leq j \leq k} p_j$. Then $p_k(x) \leq \tilde p_k(x)$, but the reverse inequality is true for balls: $B_{\tilde p_k}(0,r) \subseteq B_{p_k}(0,r)$. In other words, the bigger the seminorm, the smaller the corresponding balls.

That the seminorms $\tilde p_k$ form a local base now follows from the inclusions $$ \bigcap_{1\leq j \leq k} B_{p_j}(x,r) \subseteq B_{\tilde p_k}(x,r) \subseteq B_{p_k}(x,r)\,,$$ which are valid for all $x \in X$ and $r > 0$.