Weak forms of open mappings and α -topology

The following notations associated with a mapping f : (X, τ) → (Y, σ) have been introduced in [D. S. Janković: A note on mappings of extremely disconnected spaces, Acta Math. Hungar. (1985), 83-92]: where for each x ∊ X. Interrelationships between well-known openness-type properties (a.o.S., w.o., -openness, a.o.W.) for f, f_*, f^α, f_α are studied.