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Can we check Compactness/Connectedness of sets in $R^{2}$ using some mathematica code ?

If Yes , Can anyone tell how ?

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    $\begingroup$ How are the sets going to be specified? Mathematica has some functions such as CylindricalDecomposition that might help you, but they work primarily with inequalities in terms of $x$ and $y$. $\endgroup$ – Michael Seifert Apr 13 '16 at 14:37
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    $\begingroup$ This question is very general and the answer is "of course, we can", if you are interested in a certain topological technique related to the general problem see e.g. this anser Finding minimal distance between two surfaces $\endgroup$ – Artes Apr 13 '16 at 15:17
  • $\begingroup$ Can you give a specify example to compute $\endgroup$ – M.R. Apr 13 '16 at 19:23

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