×

On sets of points of semicontinuity in fine topologies generated by an ideal. (English) Zbl 0716.54011

Let T be a fine topology generated by an ideal on a metric space X and let f: (X,T)\(\to R\) be a function. Let us put \(C(f)=\{x:\limsup_{y\to x}f(y)\leq f(x)\leq \liminf_{y\to x}f(y)\},\) \(S^+(f)=\{x:\limsup_{y\to x}f(y)\leq f(x)\},\) \(T^+(f)=\{x:\limsup_{y\to x}f(x)<f(x)\) and \(x\) is not isolated}, \(S^-(f)=\{x:\liminf_{y\to x}f(y)\geq f(x)\},\) \(T^-(f)=\{x: \liminf_{y\to x}f(y)>f(x)\) and x is not isolated} and \(CH(f)=(C(f),S^+(f),T^+(f),S^-(f),T^-(f))\). A complete characterization is given of such quintuples of subsets of X which are of the type CH(f) for some \(f:X\to R\) whenever T belongs to a fairly large subclass of the class of all fine topologies generated by ideals.
Reviewer: Z.Grande

MSC:

54C30 Real-valued functions in general topology
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
PDFBibTeX XMLCite
Full Text: DOI EuDML