Hausdorff Property

Definition (Hausdorff Property)

A space $(X,\tau )$ is Hausdorff if $\forall x\ne y$ in $X$, there exist open sets $U,V$ where $x\in U,y\in V$ which are disjoint.

We say $U,V$ are “housed off” from one another.

Theorem (Metric Spaces are Hausdorff)

Any Metric Space is Hausdorff.

Proof: For any $x\ne y\in X$, then $0<d(x,y)$. But then we can define

$$B_{d(x,y)/3}(x)B_{d(x,y)/3}(y)$$

such that

$$B_{d(x,y)/3}(x)\cap B_{d(x,y)/3}(y)=\varnothing$$

If $\exists z$ is the intersection, the triangle inequality would be contradicted.

Theorem (Unique Limit in Hausdorff)

In a Hausdorff space, a Topological Sequence can have at most one limit.

Proof. If $(x_n)\to x$ and $y$ where $x\ne y$, we can separate them by open $x\in U,y\in V$ such that $U\cap V=\varnothing$. We can then find $N,M$ such that

$$\forall n\ge N,x_n\in U\forall m\ge M,x_m\in V$$

which is a contradiction when $n>\max (N,M)$.