Discontinuities of A-continuous functions
by
Olexandr Maslyuchenko
Chernivtsi National University

A map A:X ∍ x→ A_x ∈ 2^2X\{∅} is called a structure on a set X. As expected, a structure A on X is called topological if for every x ∈ X the following four axioms are satisfied:

(N₁) x ∈ A for every A ∈ A_x;

(N₂) if A ∈ A_x and B ⊇ A, then B ∈ A_x;

(N₃) if A, B ∈ A_x then A∩B ∈ A_x;

(N₄) for any A ∈ A_x there is B ∈ A_x such that A ∈ A_y for all y ∈ B.

A structure A is called quasitopological (pretopological) if the conditions N₁, N₂ ( and N₃) hold.

Let A be a structure on a set X and Y be a topological space. A map f:X→ Y is defined to be A-continuous at a point x ∈ X if for every neighborhood V of the point f(x) there is A ∈ A_x such that f(A) ⊆ V (a similar notion has been considered in [1]). We say that f is A-continuous if it is A-continuous at each point x ∈ X. Each structure A generates the quasitopological structure B_x={B ⊆ X: x ∈ B and A ⊂ B for some A ∈ A_x} so that the A-continuity is equivalent to the B-continuity. The elements of the system B_x are called A-neighborhoods of x. A set E ⊆ X is called A-joint if there are sets A and B such that cl(A)∩B=A∩cl(B)=∅, cl(A)∩cl(B)=E and cl(A) is an A-neighborhood of each point x ∈ E. We say that E is s-A-joint if there is a sequence of A-joint sets E_n such that E=∪_n=1^∞ E_n.

Theorem 1 Let X be a metrizable space, Y be a separable metrizable space, A be a structure on X and f:X→ Y be a quasicontinuous A-continuous map of the first Baire class. The the set D(f) of discontinuity points of f is s-A-joint.

Theorem 2 Let X be a metrizable space, A be a pretopological structure on X and E ⊆ X be a s-A-joint set. Then there is a lower semicontinuous quasicontinuous A-continuous function f:X→ R such that D(f)=E.

Theorem 3 Let X be a metrizable space, A be a pretopological structure on X. For a subset E ⊆ X the following conditions are equivalent:

(i) E is s-A-joint;

(ii) there is a semicontinuous quasicontinuous A-continuous function f:X→ R with D(f)=E;

(ii) there is a quasicontinuous A-continuous first Baire class function f:X→ R with D(f)=E.

References.

[1] V.V.Nesterenko, The set of points of a-continuity // Naukovyi Visnyk Cherniv. Univ. 269. Mathematics. - Chernivtsi: Ruta, 2005. P.79-80 (in Ukrainian).

Date received: May 7, 2008