Theorem 1.36 in Rudin's Functional Analysis

Question

I have a few questions regarding the proof of Theorem 1.36 in Rudin's Functional Analysis:

• Why does $V$ being open imply $x/t\in V$ for some $t<1$.
• How does the inequality $\mu_V(x-y)<r$ come about from $x-y\in rV$ and (a)? All I can gather is that $\mu_V(x-y)\leq r$ (note the lesser or equal) from the fact that $x-y\in rV$ and the definition of $\mu_V$.
• Why is every non-zero $x$ not in some $V$?

Answer 1

• Answer to 1): because $\{t∈\Bbb R^∗∣x/t∈V\}$ is open and contains $1.$
• Answer to 2): $(x−y)/r∈V$ implies $μ_V((x−y)/r)<1,$ i.e. $μ_V(x−y)<r.$
• Answer to 3): by Rudin's definition, all topological vector spaces are Hausdorff.