Hausdorff Axioms

Describe subsets of elements $x$ in a Neighborhood Set $E$ of $x$. The Neighborhood is assumed to satisfy:

1. There corresponds to each point $x$ at least one Neighborhood $U(x)$, and each Neighborhood $U(x)$ contains the point $x$.

2. If $U(x)$ and $V(x)$ are two Neighborhoods of the same point $x$, there must exist a Neighborhood $W(x)$ that is a subset of both.

3. If the point $y$ lies in $U(x)$, there must exist a Neighborhood $U(y)$ that is a Subset of $U(x)$.

4. For two different points $x$ and $y$, there are two corresponding Neighborhoods $U(x)$ and $U(y)$ with no points in common.