Theorem 1.27 in Rudin's Functional Analysis

Question

1.27 Theorem: Suppose $Y$ is a subspace of a topological vector space $X$, and $Y$ is an $F$-space (in the topology inherited from $X$). Then $Y$ is a closed subspace of $X$.

Here is Rudin's proof:

I don't see why $y_0$ being in $\overline{x + W}$ for every open nbhd $W$ of $0$ implies that $y_0 = x$. Would someone mind explaining this?

Answer 1

Isn't this just the Hausdorff property? If $y_0 \not= x$ there would be two neighborhoods $W_1$ and $W_2$ of $0$ with the property that $(y_0 + W_1) \cap (x + W_2) = \emptyset$. This forces $y_0 \notin \overline{x + W_2}$.