## Weak forms of openness based upon denseness.(English)Zbl 0869.54019

Z. Frolík [Czech. Math. J. 11(86), 381-385 (1961; Zbl 0104.17204)] introduced feebly open functions which are surjections that inversely preserve dense sets. K. R. Gentry and H. B. Hoyle III [ibid. 21(96), 5-12 (1971; Zbl 0222.54010)] dropped the surjectiveness condition and called the function somewhat open. D. S. Janković and Ch. Konstadilaki-Savvopoulou [Math. Bohem. 117, No. 3, 259-270 (1992; Zbl 0802.54005)] introduced the notion of nearly feebly open functions which are characterized by having dense inverse images of open dense sets.
In this paper the author defines a function $$f:X\to Y$$ to be hardly open provided that for each dense subset $$A$$ of $$Y$$ that is contained in a proper open subset, $$f^{-1}(A)$$ is dense in $$X$$. The author gives examples to show hardly open is between somewhat openness and near feeble openness. The author characterizes surjective functions which are hardly open, semicontinuous surjections which are hardly open, and gives some additional properties of hardly open functions.

### MSC:

 54C10 Special maps on topological spaces (open, closed, perfect, etc.)

### Citations:

Zbl 0104.17204; Zbl 0222.54010; Zbl 0802.54005