Hausdorff space

• Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology.

• 定義：separated by neightborhood
  • Points $x,y$ in a topological space $X$ can be separated by neightborhoods if there exists a neighborhood $U$ of $x$ and a neightborhood $V$ of $y$ such that $U, V$ are disjoint set. ($U \cap V = \phi$)
  • 拓樸空間 $X$中的任意兩點 $x,y$的對應鄰域為 $U, V$，若存在不交的兩個鄰域時，則稱 $x,y$ 可由鄰域分離。
  • 此定義中不要求為開鄰域或是閉鄰域
  • E.g. $A=[0,1), B=(1,2]$, let $U=(-1,1), V=(1,3)$.
• 定義：Hausdorff space
  • $X$ is a Hausdorff space if all distinct points in $X$ are pairwise neighborhood-spearable.

• 對於topologically space $X$ 以下敘述等價：

  • $X$ is a Hausdorff space.
  • Limits of nets, filfters in $X$ are unique.
  • $\lbrace (x,x) | x \in X \rbrace$ 為積空間 $X \times X$的閉集合。
  • Any singleton set $\lbrace x \rbrace \subset X$ is equal to the intersection of all closed neighborhoods of $x$.

• 幾乎所有在分析中的空間都是Hausdorff space. (esp. the real numbers under the standard metric topology on real numbers).

• 所有的metric spaces都是Hausdorff space.

• Pesudometric spaces不是Hausdorff space.

  • $\forall x, y, z in X$, the pseudometric $d: (x,y) \rightarrow \mathbb{R}$ satisfies:
    • $d(x,x) = 0$.
    • $d(x,y) = d(y,x)$
    • $d(x,z) \leq d(x,y) + d(y,z)$.

  • pseudometric 與 metric的差異只有在允許相異的元素距離為0，即 $x \neq y, d(x,y) = 0$.

  • E.g. Functional space $F(x) with f: X \rightarrow \mathbb{R}$, $\forall f,g \in F(x), \ x_0 in X, \ d(f,g) = | f(x_0) - g(x_0) |$.