%0 Journal Article
%T On the pointfree counterpart of the local definition of classical continuous maps
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Banaschewski, Bernhard
%D 2018
%\ 01/01/2018
%V 8
%N 1
%P 1-8
%! On the pointfree counterpart of the local definition of classical continuous maps
%K Pointfree topology
%K continuous map
%K localic maps
%R 10.29252/cgasa.8.1.1
%X The familiar classical result that a continuous map from a space $X$ to a space $Y$ can be defined by giving continuous maps $\varphi_U: U \to Y$ on each member $U$ of an open cover ${\mathfrak C}$ of $X$ such that $\varphi_U\mid U \cap V = \varphi_V \mid U \cap V$ for all $U,V \in {\mathfrak C}$ was recently shown to have an exact analogue in pointfree topology, and the same was done for the familiar classical counterpart concerning finite closed covers of a space $X$ (Picado and Pultr [4]). This note presents alternative proofs of these pointfree results which differ from those of [4] by treating the issue in terms of frame homomorphisms while the latter deals with the dual situation concerning localic maps. A notable advantage of the present approach is that it also provides proofs of the analogous results for some significant variants of frames which are not covered by the localic arguments.
%U https://cgasa.sbu.ac.ir/article_32712_7102051b8b0d2b0555b4ab6cee021fc7.pdf