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

Document Type: Research Paper

Author

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.

Keywords


[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.