Compact ∧ $k_1$ -Hausdorff ⇒ $T_2$

Immediate from the definitions.

The converse ( $T_2$ ⇒ Compact ∧ $k_1$ -Hausdorff ) cannot be proven from other theorems or disproven from a counterexample.

Compact ∧ k1k_1k1​-Hausdorff ⇒ T2T_2T2​