Analyzing a topological space by its partitions

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$\newcommand{\C}{\mathscr{C}}$Let $$ \C := \{ \frac{1}{n} \mid n>0 \} \cup \{0\} \leq \mathbb{R}^2 $$ Be the “convergent sequence space”. Let $S\subseteq C$. We have the following criteria:

$S$ is clopen if it is (co)-finite.
$S$ is open if $0\not\in S$ or $S$ is cofinite (hence contains an $\epsilon$-ball around 0).
Infinite, contains 0, not cofinite: Closed, not open.
infinite, does not contain 0, not cofinite: Open, not closed.

Now consider a finite partition $\Pi = \bigcup_{i=0}^n \Pi_i$, where wlog $0\in \Pi_0$. In every case, $\Pi_0$ is closed, since its complement consists of points not containing the $0$.

Possibility 1

$\Pi_0$ is finite.

We then know that $\Pi_0$ is not open, since it cannot contain an $\epsilon$-ball around $0$. Let wlog $\Pi_1, \ldots,\Pi_k$ be all the infinite sets. All $\Pi_i$ with $i>k$ must be clopen.

a) There is precisely one infinite set $\Pi_1$.

Since it is cofinite, it is clopen. In our quotient space $\C / \Pi$, We now look for the smallest superset of ${Pi_0} being open: This is already achieved by ${\Pi_0,\Pi_1}$, since the preimage under their

b) There is more than one infinite set.

$Q$: Which of those partitions are $T_0$? $T_1$? $T_2$?