Partition Topology is T5

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Theorem

Let $S$ be a set and let $\mathcal P$ be a partition on $S$ which is not the (trivial) partition of singletons.

Let $T = \left({S, \tau}\right)$ be the partition space whose basis is $\mathcal P$.


Then $T$ is a $T_5$ space.


Corollary

$T$ is a $T_4$ space.


Proof

Let $A$ and $B$ be subsets of $S$ such that $A^- \cap B = A \cap B^- = \varnothing$.

From Open Set in Partition Topology is also Closed, we get that $A^-$ and $B^-$ are both clopen.

From Set is Subset of its Topological Closure:

$A \subseteq A^-$

From $A^- \cap B = \varnothing$ it follows from Intersection with Complement is Empty iff Subset that $B \subseteq S \setminus \left({A^-}\right)$.

Since $A^-$ is clopen, it follows that $A^-$ and $S \setminus \left({A^-}\right)$ are both open.

Also, they are disjoint by definition.

Thus we have $A^-$, an open set containing $A$, and $S \setminus \left({A^-}\right)$, an open set containing $B$.

Thus, by definition, $T$ is a $T_5$ space.

$\blacksquare$


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