Summary: We define and study new weak versions of the classical star-Rothberger covering property using \(\alpha\)-open and \(\theta\)-open sets of a topological space. We discuss their relations with some known weak version of the Rothberger property. It is also proved that for an extremally disconnected \(S\)-paracompact-\(T_2\) space the properties: Rothberger, cf. F. Rothberger, Fundam. Math. 30, 50–55 (1938; Zbl 0018.24701)], M. Scheepers [Topology Appl. 69, No. 1, 31–62 (1996; Zbl 0848.54018)], semi-Rothberger, cf. A. Sabah and M. ud D. Khan [Bull. Iran. Math. Soc. 43, No. 6, 1969–1987 (2017; Zbl 1406.54013)], \(\alpha\)-Rothberger, cf. L. D. R. Kočinac [Filomat 33, No. 5, 1485–1493 (2019; Zbl 1499.54107)], \(\theta\)-Rothberger, cf. [loc. cit.] are equivalent. Moreover, for an extremally disconnected space the \(\theta\)-Rothberger property coincides with the almost Rothberger property, cf. M. Scheepers [Proc. Am. Math. Soc. 127, No. 1, 251–257 (1999; Zbl 0910.90287)].