  In this note we prove that a four-dimensional compact oriented
half-confor\-mally flat Riemannian manifold $M^4$ is topologically
$\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2},$ provided that the sectional
curvatures all lie in the interval $[\frac{3\sqrt{3}-5}{4},\,1].$ In addition,
we use the notion of biorthogonal (sectional) curvature to obtain a pinching
condition which guarantees that a four-dimensional compact manifold is
homeomorphic to a connected sum of copies of the complex projective plane or
the $4$-sphere.
