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).

令\({\mathcal {P}}\)是与类别\({\mathcal {C}}\)相关的子对象的属性。一个对象\(X\in {\mathcal {C}}\)是\({\mathcal {P}}\)分隔的，如果\(X\times X\) 中的对角线具有\({\mathcal {P }}\) ; 因此，例如拓扑空间（或场所）范畴中的封闭性导致豪斯多夫（或强豪斯多夫）公理。在本文中，我们研究了对角线拟合的区域设置（框架）（即，开放子区域设置的交集——我们谈论的是\({\mathcal {F}}\)分隔的区域设置）。回想一下语言环境是合适的如果它的每个子区域都适合。由于此属性由产品和子语言环境继承，因此适应度意味着 ( \({\mathcal {F}}\) sep) 被证明是严格弱的（本文的结果之一）。我们证明 ( \({\mathcal {F}}\) sep) 与强豪斯多夫公理 (sH) 平行：（1）它的特征在于系统组合结构的 Dowker-Strauss 类型属性框架同态\(L\rightarrow M\)（因此，特别地，它暗示\((T_U)\)出于类似的原因，如 (sH) 确实如此），以及 (2) 在与 (sH) 的某种对偶性中在L 中以所有几乎同态为特征（具有略微放松的加入要求的框架同态）\(L\rightarrow M\)是框架同态（而一个具有弱同态的 (sH) 特征，其中满足要求是放松的）。