Proof that the set $ \tilde{D}=\{ (x, y) \in V \times V \mid[x, y] \subseteq D \}$ is open in $V \times V$

Question

Let $D$ be an open subset of a finite-dimensional real vector space $V$ and $$ \tilde{D}=\{ (x, y) \in V \times V \mid[x, y] \subseteq D \}. $$ I would like to show that $\tilde{D}$ is open in $V \times V$.

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My proof was wrong. I am working on a new version. Thanks for hints.

Answer 1

$[u, v] \subseteq U_x \times B_y(r) \subseteq D$ is doubly incorrect, and $U_x \times B_y(r) \subseteq \tilde{D}$ is generally false. Here is a sketch of an alternative proof:

Since the compact segment $[x,y]$ is disjoint from the closed subset $D^\complement,$ their distance $r$ is positive. $B_x(r)\times B_y(r)\subseteq\tilde D.$