Open set in $\mathbb{R}^4$

Question

I have to show that the following set is open:

$$U := \{(a,b,c,d)∈\Bbb{R}^4 : |ad-bc|>1\}$$ I know I have to show that there is an open ball at every point in the set, but don't know how to show this.

Have you tried to show this? Where do you get stuck? Can you find open balls around specific points, like for example only those where $b = 0$ or $c = 0$? Alternatively, are you familiar with "the inverse image of an open set under a continuous function is open"? — Mees de Vries, Oct 16 '18 at 10:17
Would not it be easier to show that U is inverse image of an open set (with $f$ continuous)? — Picaud Vincent, Oct 16 '18 at 10:17

score 3 · Accepted Answer · answered Oct 16 '18 at 10:17

Tha map $f\colon \mathbb{R}^4 \to \mathbb{R}: (a,b,c,d) \mapsto \lvert ad-bc \rvert$ is continuous, because it is a composition of $\lvert {}\cdot{} \rvert$, which is continuous, and a polynomial expression. Hence $$f^{-1}((1,+\infty)) = \{(a,b,c,d) \in \mathbb{R}^4: \lvert ad-bc \rvert > 1 \}$$ is open.

Open set in $\mathbb{R}^4$

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