Indiscrete topological space is second countable.

Question

Is uncountable indiscrete topological space (X, tau )is second countale ? As far I know the only possible basis for the indiscrete topological space is X but since X is uncountable so its not second countable. But the basis {X} has only one element that is X . Can we write set X in the basis as {X}. If we consider this then space is second countable. Which perspective is correct ?

Answer 1

Second countability requires a countable basis for the topology -- the countability or uncountability of the space is not part of the definition. Further, it doesn't require that any particular basis be countable, only that somehow there is a countable basis.

In the indiscrete topology, $\tau = \{\varnothing, X\}$. Note that $\tau$ is a (more than sufficient) basis for $\tau$ and $\tau$ is countable, so this topology is second countable.