Conclusion about cardinalty. Assume that:
$$\left| T \right| > {\aleph _0}$$
Why can't one assume immediately that:
$$\left| T \right| \cdot \left| T \right| > \left| T \right| \cdot {\aleph _0}$$ 

For infinite cardinals $\kappa, \lambda$, we have:
$$
\kappa \times \lambda = \max(\kappa, \lambda) \text{ and, therefore } \\
\kappa \times \kappa = \kappa \text{ for $\kappa$ infinite}
$$


In this case, $|T|\cdot |T| = |T|$. Also, $|T|\cdot \aleph_0 = \max(|T|,\aleph_0) = |T|$. So in particular, $|T|\cdot |T| \not > |T|\cdot \aleph_0$, $|T|\cdot |T| = |T|\cdot \aleph_0$, 
