# Under what conditions the set $A$ is connected? [closed]

Let $f:ℝ^{m}→ℝ$ is a continuous function. Let us consider the set $$A=\{x∈ℝ^{m} \ |\ f(x)=0\} = f^{-1}[\{0\}]$$

My question is: Under what conditions the set $A$ is connected?, simply connected?

## closed as too broad by user99914, Moishe Kohan, user91500, Qiaochu Yuan, user370967 Sep 23 '17 at 15:32

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• Convexity might be relevant here. Not sure though. – Shalop Sep 22 '17 at 9:28
• The exact conditions are of course whenever $f$ has the property that $f^{-1}(0)$ is connected/simply connected. I doubt there is some other meaningful property of $f$ that is equivalent to that... – skyking Sep 22 '17 at 9:31
• Convexity of $f$ is not enough. For example $f(x)=x^2-2$ would be counterexample. In higher dimensions, this example does not work though. – humanStampedist Sep 22 '17 at 9:32
• Not sure if we can conclude (any topological properties of $A$) if $m > 1$ and if $f$ isn't bijective, if I'm wrong I'm really interested to see a counter example – Perturbative Sep 22 '17 at 9:45