There is no such topology.

Suppose there were. Then $\mathbb R$ itself is not compact, so there is an open cover $\mathcal U$ of $\mathbb R$ with no finite subcover. Using recursion, we can construct a non-compact countable subset of $\mathbb R$.

To begin, let $x_0 \in \mathbb R$ and let $U_0$ be any member of $\mathcal U$ containing $x_0$. Since $\{U_0\}$ does not cover $\mathbb R$, we may choose $x_1 \in \mathbb R - U_0$, and then we may also choose some $U_1 \in \mathcal U$ containing $x_1$. Continuing in this way, suppose $x_0, \dots, x_n$ are already chosen, and so are sets $U_0, \dots, U_n$ from $\mathcal U$, with $x_i \in U_i$ for each $i$. Since $\{U_0, \dots, U_n\}$ does not cover $\mathbb R$, we may choose $x_{n+1} \in \mathbb R - \bigcup_{i \leq n}U_i$, and then we may also choose $U_{n+1}$ from $\mathcal U$ containing $x_{n+1}$.

In the end, we get a countable set $\{x_0,x_1,x_2,\dots\}$ and a countable open cover $\{U_0,U_1,U_2,\dots\}$ of this set. But it is clear from our choices of the $x_i$ and the $U_i$ that this open cover has no finite subcover.