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

Document Type : Research Paper


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



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.


[1] Banaschewski, B., Another look at the localic Tychono theorem, Comment. Math. Univ. Carolinae 29 (1988), 647-656.
[2] Mac Lane,  S., ``Categories for the Working Mathematician", Springer-Verlag, 1971.
[3] Picado, J. and Pultr, A., ``Frames and Locales: Topology without Points", Frontiers in Mathematics 28, Springer, 2013.
[4] Picado, J. and Pultr, A., Localic maps constructed from open and closed parts, Categ. General Alg. Struct. Appl. 6(1) (2017), 21-35.