which of these is a Hausdorff Space?

Question

I have found the topologies on the set $\{a, b\}$:

$\{\emptyset, \{a, b\}\}$,

$\{\emptyset, \{a\}, \{a, b\}\}$,

$\{\emptyset, \{b\}, \{a, b\}\}$

and $\{\emptyset, \{a\}, \{b\}, \{a, b\}\}$,

and now want to determine which of these is Hausdorff/separated.

I have the definition that in order of for this each distinct point of the space must have a disjoint neighbourhood.

My thinking is that the only separated space in the list is $\{\emptyset, \{a, b\}\}$, because it is the only one that has no two elements with both $a$ and $b$?

Answer 1

Actually, that one is not separated: there are no neighborhoods of $a$ and $b$ with empty intersection. And the same thing occurs with $\bigl\{\emptyset,\{a\},\{a,b\}\bigr\}$ and with $\bigl\{\emptyset,\{b\},\{a,b\}\bigr\}$.

On the other hand, in the case of $\bigl\{\emptyset,\{a\},\{b\},\{a,b\}\bigr\}$ you do get a Hausdorff space: $\{a\}$ is a neighborhood of $a$, $\{b\}$ is a neighborhood of $b$, and $\{a\}\cap\{b\}=\emptyset$.

Answer 2

You got it the other way around: to be Hausdorff, each couple of points of the space must have two disjoint open neighborhoods (one neighborhood for each point!). This is what gives a Hausdorff space its "separatedness", two points can always be separated by two disjoint neighborhoods.

In your example, the only topology which is Hausdorff is the discrete topology $\{\varnothing, \{a\},\{b\},\{a,b\}\}$, because $a$ and $b$ need to be separated, and you can do this only if both $\{a\}$ and $\{b\}$ (the only "plausible" neighborhoods of the two points) are open sets.

$\{a,b\}$ would also be a neighborhood for both $a$ and $b$, yes, but it doesn't separate them. Similarly, if $\{a\}$ is open and $\{b\}$ isn't, you can't separate $a$ and $b$ because $\{a\}$ and $\{a,b\}$ aren't disjoint neighborhoods.

Answer 3

A space $X$ is Hausdorff if, given $x,y\in X$, with $x\ne y$, there are open sets $U,V$ such that $x\in U,y\notin U$ and $x\notin V,y\in V$.

You need a pair of open sets for any pair of distinct points.

By the way, a finite set is Hausdorff if and only if the topology is discrete, because in a Hausdorff space singleton sets are closed.