[Math] Is $\mathbb{R^2}$ Hausdorff? Give an example of a non-Hausdorff topology on $\mathbb{R}$

Question

these are two questions on Hausdorff topological spaces. The bit I am having particular difficulty with is finding an 'example of a non-Hausdorff topology on $\mathbb{R}$'

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A Hausdorff topological space $(X, \tau)$ is such that any distinct points $a, v \in X$ have disjoint open neighbourhoods. i.e. there are open neighbourhoods $U_a, V_b \in \tau$ such that $ a \in U_a $ and $b \in V_b $ and $U_a \cap V_b = \emptyset$

Is $\mathbb{R^2}$ Hausdorff?

I believe so. Take $a, b \in \mathbb{R^2}$. Take open neighbourhoods:

$U_a=B_{r_a}(a)=\{(x, y) : |(x, y)-a|<r_a\}$

$V_b=B_{r_b}(b)=\{(x, y) : |(x, y)-b|<r_b\}$

Let $r=d(a,b)$. Take $r_a=r_b=\frac{r}{2}$

So $\mathbb{R^2}$ is Hausdorff.

Is this correct?

Give an example of a non-Hausdorff topology on the set of real numbers

$\mathbb{R}$ is clearly Hausdorff. What is another example of a topology on $\mathbb{R}$ ?

Please could you help me with this one?

Answer 1

There are a few "natural" examples, e.g.:

• the trivial topology already mentioned in Mandrathax answer;
• the cofinite topology, where the closed sets are precisely $\Bbb R$, $\emptyset$ and all finite subsets;
• the topology whose non-trivial open sets are the right halflines $(a,\infty)$;
• as the latter but with the left halflines $(-\infty,b)$.