( $T_1$ ∧ First Countable ∧ $k$-Menger ) ⇒ Hemicompact

See Proposition 5 of DOI 10.1016/j.topol.2005.07.015. In that proof, one can take $O_n(K)$ to be $X \setminus \{ x_n(K) \}$, which is open by $T_1$.

• Converse
• References

The converse ( Hemicompact ⇒ ( $T_1$ ∧ First Countable ∧ $k$-Menger )) cannot be proven or disproven from other known theorems