Lie Groups of bigger cardinality

Question

A Lie Group is to be a group that is also a manifold, and of course a manifold is a second countable Hausdorff space. Now the maximum cardinality for a second countable (Hausdorff)space is $\beth_{2}$ = $\mathcal P^{2} \left({\mathbb{N}}\right)$. Now most of the classical Lie groups have the cardinality $\beth_{1}$(the cardinality of the continuum). Are there any examples of Lie groups with the largest possible cardinality? If there please expound on them.

Answer 1

A second countable $T_1$ space has cardinality at most $\mathcal P(\Bbb N)$. For let $\tau_X$ be its topology; we may write any open set as a union of elements of the countable basis. Picking such a decomposition for each open set gives us an injection $\tau_X \hookrightarrow \mathcal P(\Bbb N)$. Because our space is $T_1$, points are closed, and hence we also have an injection $X \hookrightarrow \tau_X$ (given by $x \mapsto X \setminus \{x\}$). So not only is the cardinality of $X$ at most that of $\mathcal P(\Bbb N)$, the cardinality of the topology is, too. Of course, Hausdorff space, hence manifolds, are $T_1$. (As a result of this and the locally Euclidean property, positive dimensional manifolds have cardinality $\big|\mathcal P(\Bbb N)\big|$.)