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Consider $W = \{x \in \mathbb{R}^2 : x_1^2 + x_2^2 < 4, x_1 \geq 1, x_2 \geq -0.5\}$

Is $W$ closed? Is $W$ convex? Is $W$ open? Is $W$ bounded? Explain your answers.

I'm a bit puzzled when I encounter such questions. I don't know how you easily see if a set like the one above is convex, open, closed etc. Can anybody help me?

Community
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Colt
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1 Answers1

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Hints :

Is it closed ? Consider the series $x_n = (2 - \frac{1}{n},0)$

Is it convex? Is the intersection of convex spaces convex?

Is it open? Consider the point $(1,0)$ and its neighbourhoods

Is it bounded? Can you find a point $x$ in $W$ such that $||x||>2$? Are all norms equivalent in $\mathbb{R}^2$?

Vincent
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