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There is a separation axiom between $T_1$ and $T_2$ that Aull (Separation of bicompact sets, Math. Annalen 158 (1965), 197–202) calls $J_1$ and that Mukherji (On weak Hausdorff spaces, Bull. Calcutta Math. Soc. 58 (1966), 153–157) calls $T_2'$; namely, "every compact subspace is closed." Such spaces are sometimes called "weak Hausdorff spaces," although this term is sometimes used to mean something else. See Hoffmann (On weak Hausdorff spaces, Arch. Math. (Basel) 32 (1979), 487–504) for further discussion of other separation axioms between $T_1$ and $T_2$.