Banaschewski, B. (2018). On the pointfree counterpart of the local definition of classical continuous maps. Categories and General Algebraic Structures with Applications, 8(1), 1-8.

Bernhard Banaschewski. "On the pointfree counterpart of the local definition of classical continuous maps". Categories and General Algebraic Structures with Applications, 8, 1, 2018, 1-8.

Banaschewski, B. (2018). 'On the pointfree counterpart of the local definition of classical continuous maps', Categories and General Algebraic Structures with Applications, 8(1), pp. 1-8.

Banaschewski, B. On the pointfree counterpart of the local definition of classical continuous maps. Categories and General Algebraic Structures with Applications, 2018; 8(1): 1-8.

On the pointfree counterpart of the local definition of classical continuous maps

^{}Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada.

Abstract

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.