# Topological invariance of intersection number

Let $X$ and $Y$ be algebraic curves in ${\mathbb P}^2$ and suppose they have an isolated intersection at $(0,0)$. Let $\hat{X}$ and $\hat{Y}$ be another such pair, and suppose that there exist neighborhoods $U$ and $\hat{U}$ of $(0,0)$ and a homeomorphism $\phi:U\rightarrow\hat{U}$ such that $\phi(U\cap X)=\hat{U}\cap \hat{X}$ and $\phi(U\cap Y)=\hat{U}\cap \hat{Y}$. It seems to be well-known that the pairs $X,Y$ and $\hat{X},\hat{Y}$ have the same intersection number at $(0,0)$. Who first proved this, and how? Where might an accessible modern presentation be found?

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it may be helpful to know that the 'reason' this number is a homeomorphism invariant is that it's equal to the linking number of the links of $X$ and $Y$ (i.e. you intersect them with the boundary of a small ball around $(0,0)$). – Vivek Shende Feb 7 '13 at 4:35
Have a look at Brieskorn's book Plane Algebraic Curves. He has a lot historical data in that book and you might find the answer there. – Liviu Nicolaescu Feb 7 '13 at 10:21