Hausdorff spaces

In the preceding post (coffee break 8-2), a topological space is defined by open subsets.

Putting simply, a topological space is a collection of open sets.

However, as this definition is too general, some problems will occur.

First of all, when $a_n\rightarrow x$  and $a_n\rightarrow y$ , $x=y$  may not be proved.

$a_n\rightarrow x$  means that, for an $N<\infty$ , if $N<n$ , all $a_n$ is included

in the neighborhood of $x$ , where the neighborhood of $x$  is an open subset
in the topological space which includes $x$ .

In the definition of a topological space, we may not say the neighborhood of $x$ and $y$  is
same or different.

Therefore, we prepare the topological space such that, if two elements $x$  and $y$  is different,

each neighborhood of elements is pairwise disjoint. Such a space is called Hausdorff space,
a separable space, or $T2$ space.