I'm trying to understand the proof of Schauder-Tychonoff fixed point theorem on page $96-97$, in Fixed Point Theory and Applications, Ravi P. Agarwal,Maria Meehan,Donal O'Regan, which can be found here in googlebooks.
Let $F: C \to E$ be a continuous function from $C$ which is a convex subset of $E$, to $E$. $E$ is Hausdorff space, so $C$ is automatically Hausdorff. I'm confused with the following statement on page $97$.
I'm not sure what $V_x(x)$ and $W_x(F(x))$ actually are. One guess is that they're just neighbourhoods of $x$ and $F(x)$ respectively. The other is that they're both neighbourhoods of $0$ at first place, and they also contain $x$ and $F(x)$ respectively. But either way I see no need to invoke Hausdorff condition to guarentee the existence of the two neighbourhoods. Because we can just let $W_x(F(x)) = E$ and then $(8.5)$ and $(8.6)$ will always hold. So what is role of Hausdorff condition here?
$E$ is a topological vector space. So the topology is translation-invariant, for any $x \in E$, the neighbourhoods of $x$ are the sets of the form $x + V = \{x + v \colon v \in V\}$ where $V$ is a neighbourhood of $0$.
$V(x)$ is just another notation for $x + V$, so for $V_x$ a neighbourhood of $0$, $V_x(x)$ is indeed a neighbourhood of $x$..
So by $x \neq F(x)$ and the Hausdorff condition, you have two neighbourhoods $U_1$ of $x$ and $U_2$ of $F(x)$ with $U_1 \cap U_2 = \varnothing$.
By the continuity of $F$, for every neighbourhood $U_3$ of $F(x)$, there is a neighbourhood $U_4$ of $x$ such that $F(C\cap U_4) \subset U_3$.
Now pick $U_3 = U_2$ and $W_x = U_3 - x$, and $V_x = (U_1 \cap U_4) - x$.
I'm rather convinced the $V_x(x) \cap W_x(F(x)) \neq \varnothing$ is a typo and should be $V_x(x) \cap W_x(F(x))= \varnothing$.