I have started reading Rudin's Functional Analysis, and in Section 1.6, he makes the following definition:
A subset $E$ of a topological vector space is said to be bounded iff for every open neighborhood $V$ of $0$, there exists an $s > 0$ such that for each $t > s$, we have $E\subseteq tV$.
Question: Since the open neighborhoods of $0$ completely determine the topology here, I have a hunch that instead of open neighborhoods of $0$, we could have taken any arbitrary point of the space. However, I am having trouble proving the equivalence. Any leads?