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Let $\tau = \{U\subseteq \mathbb{N} : \mathbb{N} - U$ is finite $\}\cup \{\emptyset\}$. This defines a topology on $\mathbb{N}$.

Is $(\mathbb{N},\tau)$ a Hausdorff space?

If not, give all valid reasons why this space cannot be a Hausdorff space.

A

Yes, $(\mathbb{N},\tau)$ is a Hausdorff space.

B

No, $(\mathbb{N},\tau)$ fails to be a Hausdorff space because singleton sets need not be closed.

C

No, $(\mathbb{N},\tau)$ is not a Hausdorff space because $\tau$ is not induced by any metric.

D

No, $(\mathbb{N},\tau)$ is not a Hausdorff space because there are no nonempty disjoint open sets.

E

No, $(\mathbb{N},\tau)$ is not a Hausdorff space because there exists $x\in X$ that is not contained in any open set besides $X$.

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