# Existence of compact Hausdorff topologies

Consider the following statement

$(C_2)$: For every set $X$ there exists a compact Hausdorff topology on $X$.

$C_2$ is a theorem of $ZFC$, because Choice gives us a bijection between any infinite set and the successor ordinal of the cardinality of the set, and any successor ordinal is compact and Hausdorff in its order topology.

Question: Is $C_2$ a theorem of $ZF$?

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$ZF$ already proves $C_2$. Given a nonempty set $X$, pick a point $p$ in $X$ and declare a set to be open if it either does not contain $p$ or is cofinite. $ZF$ proves this topology to be compact and Hausdorff.
This is called "the one point compactification" of a discrete space $X \setminus \{p\}$. –  GEdgar Sep 26 '11 at 17:48