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Show that the set $S=\{a \in \mathbb{R}^3\,| \,a_1 +a_3^2 \sin(a_1+a_2)\geqslant a_3\}$ in closed in $\mathbb{R}^3$ with the euclidean metric.

I know that I would probably have to show that the boundary of $S$ is contained in $S$, but I really don't know how to actually go about this one.

Any help will be appreciated.

Francis
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johny
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  • Do you know that a function $f \colon X \to Y$ between topological spaces $X$ and $Y$ is continuous if and only if the preimages of closed sets in $Y$ are closed sets in $X$? Then take $X = \mathbb{R}^3$ and $Y = \mathbb{R}$, and find a suitable $f$. – Daniel Fischer Oct 22 '13 at 09:46
  • @DanielFischer don't really understand what you are trying to say, how would finding such an "f" help with this question ? – johny Oct 22 '13 at 09:54
  • @johny : Can you write your set as $S = {a \in \mathbb{R}^3 : f(a_1,a_2,a_3) \geq 0}$ What is $f$ there? Is it continuous? – Prahlad Vaidyanathan Oct 22 '13 at 09:56
  • @PrahladVaidyanathan Yes i can definitely write it like that,also i believe that f should be continuous. – johny Oct 22 '13 at 10:06
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    Now notice that $S = f^{-1}[0,\infty)$ and use @Daniel's comment – Prahlad Vaidyanathan Oct 22 '13 at 10:08
  • @DanielFischer I have one doubt in my mind regarding this question, what was the need to specify,closed in $R^3$ with the "Euclidean metric"? Why is it necessary for the metric to be Euclidean? – johny Oct 23 '13 at 14:01
  • The metric determines the topology, and the topology determines which functions are continuous. The function $f(a_1,a_2,a_3) = a_1 + a_3^2\sin(a_1+a_2)-a_3$ is not continuous for every metric on $\mathbb{R}^3$ (though don't ask me to give an example where it wouldn't, that would look very artificial), but it's continuous when $\mathbb{R}^3$ is topologised by the Euclidean metric. Some time later, you will refer to that topology as the "standard topology". – Daniel Fischer Oct 23 '13 at 14:27
  • @DanielFischer Alright, got it.Thank you. – johny Oct 23 '13 at 14:32

1 Answers1

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The set $S$ is closed if any converging sequence $\left\{ a_{n}\right\} $ of elements of $S$ converges to an element $a$ that also belongs to $S$.

Define $$f\left( a_{n}\right) =a_{n,1}+a_{n,3}^{2}\sin \left( a_{n,1}+a_{n,2}\right) -a_{n,3}$$ so that $$S=\left\{ s:f\left( s\right) \geq 0\right\}$$

Since limits preserve weak inequalities, we have that $$\lim_{n\rightarrow \infty }f\left( a_{n}\right) \geq 0$$

Since $f$ is continuous, we have that $$\lim_{n\rightarrow \infty }f\left( a_{n}\right) =f\left( \lim_{n\rightarrow\infty }a_{n}\right) =f\left( a\right)$$

Therefore, also $a$ belongs to $S$ because it satisfies the weak inequality $$f\left( a\right) \geq 0$$

user4422
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