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

Bernhard Banaschewski

doi:10.29252/cgasa.8.1.1

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 $varphi_Umid U cap V = varphi_V mid U cap V$ 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.

Highlights

X to a space Y can be defined by giving continuous maps φU : U → Y on each member U of an open cover C of X such that φU | U ∩ V = φV | U ∩ V for all U, V ∈ 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])
The purpose of this note is to present an alternative to the proof given loc. cit.; while the latter uses arguments about sublocales, it will be shown here that surprisingly simple considerations involving only frame homomorphisms prove the result in question
A cover of a frame L is a subset C of L whose joint C equals e, the top of L; stands for the usual product of frames; for any element a of a frame L, ↓a = {s ∈ L | s ≤ a} which is the image of L by the frame homomorphism (·) ∧ a : L → ↓ a, s → s ∧ a, and for ↑a = {s ∈ L | s ≥ a} and (·) ∨ a : L → ↑a, s → s ∨ a

Summary

Introduction

Concerning Proposition 1.2, the desired h : M → L is obviously given by h(s) = {ha(s) | a ∈ C}, and without Lemma 1.1 one could just consider this map M → L and show (i) it is a homomorphism such that (ii) a ∧ h(s) = ha(s) for all a ∈ C and s ∈ M . Regarding the earlier comment that the arguments used here are all applicable to the κ-frames for any regular cardinal κ, it should be pointed out that, in this situation, the covers involved are those provided by the given setting, that is, the κ-small subsets C such that C = e: this trivially implies that all the other joins which enter into the argument exist.

Results

Conclusion