Some versions of the fundamental principles of the functional analysis in asymmetric
normed spaces – the Open Mapping Theorem, the Closed Graph Theorem and
the Uniform Boundedness Principle – are proved. The proofs are based on an
asymmetric version of a lemma of Zabreijko on the continuity of the
countably subadditive functionals. At the same time a flaw in the proof of the
Uniform Boundedness Principle, given in the book by Cobzaş (2013), is fixed.