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Suppose that $A,B$ are real analytic subsets of $\Omega\subseteq \mathbb{R}^n$ and $p\in A\cap B \neq \emptyset$. Does the intersection inequality from complex analysis still hold, i.e. does the following inequality hold:

$\mathrm{dim}^{\mathbb{R}}_p(A\cap B) \geq \mathrm{dim}^{\mathbb{R}}_p(A)+\mathrm{dim}^{\mathbb{R}}_p( B)-n$?

By real analytic subset, I mean sets that are locally given as the zero set of finitely manly real analytic functions and by dimension, I mean the maximal dimension of regular points near $p$ as manifolds. The complex analytic proofs that I know use the "Active Lemma", i.e. $\mathrm{dim}(\left\{f=0\right\})=n-1$, for non-zero $f$, which does not hold over $\mathbb{R}$.

If not, are there explicit counterexamples?

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  • $\begingroup$ I don't fully understand your definition of dimension, so can I get an example: if you look at the set defined in $\mathbb{R}^3$ by $z^2 - x^2 - y^2 = 0$ (so a bi-cone), what is its dimension at the origin? (is it 2?) $\endgroup$
    – Willie Wong
    Commented Mar 2, 2022 at 13:40
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    $\begingroup$ (If the dimension is 2, then you have a counterexample by intersecting the bi-cone with the plane $z = 0$ which is two dimensional.) $\endgroup$
    – Willie Wong
    Commented Mar 2, 2022 at 13:41
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    $\begingroup$ A simpler example is a sphere in three space and a plane tangent to it . $\endgroup$
    – Mohan Ramachandran
    Commented Mar 2, 2022 at 20:33
  • $\begingroup$ @MohanRamachandran you might as well post that as an answer. $\endgroup$
    – Willie Wong
    Commented Mar 2, 2022 at 21:02

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For a counterexample take a sphere in 3 space and a plane tangent to it .

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