Tameness of Complex Dimension in a Real Analytic Set Given a real analytic set $X$ in a complex manifold and a positive integer $d$, denote by $\mathcal A^d$ the set of points $p$ in $X$ at which there exists a germ of a complex analytic set of dimension $d$ contained in $X$. It is proved that $\mathcal A^d$ is a closed semianalytic subset of $X$. Keywords:complex dimension, finite type, semianalytic set, tamenessCategories:32B10, 32B20, 32C07, 32C25, 32V15, 32V40, 14P15