Studies such as Dube et al. (Topology Appl. 160(18):2454–2464 2013), Georgiou et al. (Algebra Universalis 80(2):16, 2019), Iliadis (2005; Topology Appl. 179, 99–110, 2015; Topology Appl. 201, 92–109, 2016) focus on the notion of saturated class of spaces and frames and the existence of universal elements in such classes. Recently, the notion of dimension for frames has been combined with this universality property (Georgiou et al., Algebra Universalis 80(2):16, 2019; Iliadis, Topology Appl. 201:92–109, 2016). In this paper we study such a property, combining it with the notion of the covering dimension dim for frames (Charalambous, J. London Math. Soc. 8(2):149–160, 1974). We define the base dimension like-function of the type dim for frames and, based on the notion of the saturated class of bases, which is inserted and studied in Georgiou et al. (Order 37:427–444, 2020), we prove that in a class of bases which is characterized by this dimension there exist universal elements. Also, we study the notion of saturated class of frames, which is inserted in Georgiou et al. (Algebra Universalis 80(2):16, 2019), proving that in classes of frames which are determined by the covering dimension dim there exist universal elements.

Dube 等人的研究。(Topology Appl. 160 (18):2454–2464 2013)，Georgiou 等人。(Algebra Universalis 80 (2):16, 2019), Iliadis (2005; Topology Appl. 179 , 99–110, 2015; Topology Appl. 201 , 92–109, 2016) 关注空间和框架饱和类的概念以及此类类中普遍元素的存在。最近，框架的维度概念与这种普遍性属性相结合（Georgiou 等人，Algebra Universalis 80 (2):16, 2019; Iliadis, Topology Appl. 201 :92–109, 2016）。在本文中，我们研究了这样一个属性，将其与帧的覆盖维度 dim 的概念相结合 (Charalambous, J. London Math. Soc. 8(2):149–160, 1974)。我们为框架定义了类似dim 类型的基维函数，并且基于饱和基类的概念，Georgiou 等人插入并研究了该概念。(Order 37 :427–444, 2020)，我们证明在以这个维度为特征的一类基中存在普遍元素。此外，我们研究了 Georgiou 等人中插入的饱和帧类的概念。(Algebra Universalis 80 (2):16, 2019)，证明在由覆盖维度 dim 确定的框架类中存在通用元素。