a neighborhood of an intersection point if a point $x$ is in the intersection of two spaces $X$ and $Y$ suppose we know explicitly a neighborhood of $x$ in $X$, can we take the same neighborhood of $x$ in $Y$. More specifically, if the neighborhood of $x$ in $X$ is homeomorphic to $\mathbb R^n$ can we say that if the neighborhood of $x$ in $Y$ is $\mathbb R^k$ then $k$ must be equal to $n$?
 A: Take $Z$ to be the subspace of $\mathbb{R}^3$ consisting of the union of $X=$ the x-y plane and $Y = $ the z-axis.  Take $x\in X\cap Y$ to be the origin (the only thing it can be).  Then a small open ball around $x$ (in $\mathbb{R}^3$) intersects $X$ in a small open 2-d ball and intersects $Y$ in a small open 1-d ball.
So, no, you can't conclude $k=n$.
A: If I understood well, X,Y are manifolds and x is in W=$X\cap Y$, and the manifolds intersect in the "right way" (transversally),the dimension of the intersection will be equal to the dimension of the ambient space minus the sum of the codimensions. 
Edit: After searching, I found we can say even more: the transversal intersection of submanifolds of a manifold is a submanifold; if Z is the intersection, and p is in Z, then we can show that actually, there is a coordinate chart around p in which X corresponds to $R^r x {0}$ and Y corresponds to ${0} x R^s$, (I think these are an inmersion and a submersion) so that  $Z=X \cap Y$,  corresponds to ${0} x R^k x {0}$ (where r=dim(X), s=dim(Y), and k=(r+s)).
