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Topology

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Non-sequentially Compact Spaces

TOPO-AGFTBN

Let $X=\mathcal{P}(\mathcal{P}(\mathbb{R}))$ be the power set of the power set of $\mathbb{R}$, with topology generated by,

$$\{U_r\} \cup \{V_r\}$$

where,

$U_r = \{A\in X:r\in A\}:r\in\mathcal{P}(\mathbb{R})\}$
$V_r = \{A\in X: r\not\in A\}:r\in\mathcal{P}(\mathbb{R})\}$

Which of the following is(are) TRUE? Select ALL that apply:

A

$X$ is separable.

B

$X$ is sequentially compact.

C

$X$ is Hausdorff.

D

$X$ is countable.