Abstract
Let \({\mathcal {P}}\) be a property of subobjects relevant in a category \({\mathcal {C}}\). An object \(X\in {\mathcal {C}}\) is \({\mathcal {P}}\)-separated if the diagonal in \(X\times X\) has \({\mathcal {P}}\); thus e.g. closedness in the category of topological spaces (resp. locales) induces the Hausdorff (resp. strong Hausdorff) axiom. In this paper we study the locales (frames) in which the diagonal is fitted (i.e., an intersection of open sublocales—we speak about \({\mathcal {F}}\)-separated locales). Recall that a locale is fit if each of its sublocales is fitted. Since this property is inherited by products and sublocales, fitness implies (\({\mathcal {F}}\)sep) which is shown to be strictly weaker (one of the results of this paper). We show that (\({\mathcal {F}}\)sep) is in a parallel with the strong Hausdorff axiom (sH): (1) it is characterized by a Dowker-Strauss type property of the combinatorial structure of the systems of frame homomorphisms \(L\rightarrow M\) (and therefore, in particular, it implies \((T_U)\) for analogous reasons like (sH) does), and (2) in a certain duality with (sH) it is characterized in L by all almost homomorphisms (frame homomorphisms with slightly relaxed join-requirement) \(L\rightarrow M\) being frame homomorphisms (while one has such a characteristic of (sH) with weak homomorphisms, where meet-requirement is relaxed).
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Notes
A space is sober if every completely prime filter \({\mathcal {F}}\) in \(\Omega (X)\) (that is, an \({\mathcal {F}}\) such that \(\mathop {\textstyle \bigcup }_{i\in J}U_i\in {\mathcal {F}}\) only if \(U_j\in {\mathcal {F}}\) for some \(j\in J\)) is \(\{U \, | \, x\in U\} \) for some \(x\in X\) – in other words, if every system of open sets that looks like a neighborhood system is really a neighborhood system of a point. For instance every Hausdorff space is sober.
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Acknowledgements
The authors acknowledge financial support from the Basque Government (Grant IT974-16), the Centre for Mathematics of the University of Coimbra (UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES) and the Department of Applied Mathematics (KAM) of Charles University. The first named author also acknowledges support from a predoctoral fellowship of the Basque Government (PRE-2018-1-0375).
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Communicated by Maria Manuel Clementino.
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Arrieta, I., Picado, J. & Pultr, A. A New Diagonal Separation and its Relations With the Hausdorff Property. Appl Categor Struct 30, 247–263 (2022). https://doi.org/10.1007/s10485-021-09655-9
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DOI: https://doi.org/10.1007/s10485-021-09655-9
Keywords
- Frame
- Locale
- Sublocale
- Preframe
- Preframe homomorphism
- Weak homomorphism
- Binary coproduct of frames
- Diagonal map
- Strongly Hausdorff frame
- Fit frame
- \(T_U\)-frame
- Simple extension