Counter example in topology

Question

We known that every subspace of second countable space is second countable. I was think if A is subspace of X and A is second countable space need to be the whole space is second countable space? My statements is it true? Given an example plz

Answer 1

Suppose that whenever $A$ is a second countable subspace of a space $X$, then $X$ is also second countable. This proposition is logically equivalent to its contrapositive: if a space $X$ is not second countable, and $A\subseteq X$, then $A$ is not second countable. But this statement is clearly false, because every finite space is second countable, and there are spaces that are not second countable.

For instance, if $X$ is an uncountable space with the discrete topology, then $X$ is not second countable, but every countable subset of $X$ is second countable.