Non normality of R R with the graph topology

Abstract

Author considers the subspaces of Y X which consists of almost continuous functions. (A function is almost continuous if every neighbourhood of its graph contains the graph of some continuous function. [S. A. Naimpally, Trans. Am. Math. Soc. 123, 267-272 (1966). All function spaces are provided by graph topology, which is stronger than Tikhonov's one and weaker then box one [loc. cit.].

Main results are: Theorem 9. The space of all almost continuous functions in R R whose graphs are dense in R 2 is not regular with respect to the graph topology. Theorem 11. There exists a function in R R (not continuous) that can not be separated from the closed set of all connected graphs, in graph topology. These theorems answer some questions from L. B. Lawrence, Houston J. Math. 13, 389-403 (1987). It was shown, that the space of all almost continuous functions from R to R is not connected in graph topology.