$$\left\{ {(x,y)\in R^2 : sin(\frac {x^3}{x^2+y^4}) >0}\right\}$$
In this case, I cannot make my function as preimage of open interval of continuous function.
Any suggestion on approaching this problem?
Is this set open in $R^2$ with Euclidean topology?
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general-topology
1 Answers
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Put $f(x, y) = \sin(\frac{x^3}{x^2+y^4})$. Note that $f:\mathbb{R}^2\backslash \{(0,0)\}\to \mathbb{R}$ is continous. Since your set is equal to $f^{-1}((0, \infty))$ then it is open in $\mathbb{R}^2\backslash \{(0,0)\}$. But $\mathbb{R}^2\backslash \{(0,0)\}$ is an open subset of $\mathbb{R}^2$ so $f^{-1}((0, \infty))$ being an open subset of an open subset is open in the whole space.