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# br We are now ready to

We are now ready to prove the counterpart of Proposition 6.4 for κ-collectionwise normality.

This is the pointfree counterpart of the classical result, originally proved in [23], that κ-collectionwise normality is hereditary with respect to -sets. (It may be worth emphasizing that the localic proof is much simpler.) In particular, it follows that any closed sublocale of a collectionwise normal locale is collectionwise normal.
Let be a frame homomorphism, with corresponding localic map , right adjoint to h. Recall that h is closed if for every and . Lemma 6.5 has another nice consequence:

In this section, we characterize κ-collectionwise normality in terms of continuous hedgehog-valued functions. The first theorem extends Urysohn's separation theorem for normal frames ([4], [2]), which corresponds to the particular case , as .

Our second theorem is a Tietze-type extension theorem for continuous hedgehog-valued functions. To prove it we need first to introduce some terminology and to recall, from [20], a glueing result for localic maps defined on closed sublocales (that we reformulate here in terms of frame homomorphisms).
For each sublocale S of a frame M, we say that a frame homomorphism has an extension to M if there exists a further frame homomorphism such that the diagram commutes. In that case we say that extends h.

Note that, by letting , this theorem yields Tietze's extension theorem for normal frames ([2]) as a particular case since .

An alternate description of metric hedgehogs
Given a frame L and a set I with cardinality κ consider the frame product , that is, the cartesian product ordered pointwisely, regarded as the collection of all maps ordered by if and only if for every , with projections For each and , let be given by for any . Suppose L has a point, that is, a completely prime filterF (specifically, a proper filter satisfying for some ). Given such L, κ and F let be the subframe of generated by Note that forms a 1,2-Dimyristoyl-sn-glycerol of , since it is obviously closed under finite meets.

Acknowledgements
The authors acknowledge financial support from the grant MTM2015-63608-P (MINECO/FEDER, UE), from the Basque Government (grant IT974-16 and Postdoctoral Fellowship POS_2016_2_0032) and from the Centre for Mathematics of the University of Coimbra (UID/MAT/00324/2013 funded by the Portuguese Government through FCT/MEC and by European RDF through Partnership Agreement PT2020).

Introduction
Obesity, which has been recognized as a disease by the American Medical Association (AMA) in 2013, is a medical condition in which excess body fat has accumulated to an extent that it may have a negative effect on health. Obesity has been linked to several kinds of diseases including type 2 diabetes, cardiovascular disease and even cancer [1]. Body fat mass is controlled by the balance between lipolysis and adipogenesis. Adipogenesis is a critical process that pluripotent mesenchymal stem cells (MSCs) differentiated into lipid laden, insulin-sensitive mature adipocytes [2]. In the process of transformation, preadipocytes undergo significant changes in morphology and gene expression, and numerous key stimulators have been identified, including peroxisome proliferator-activated receptor γ (PPARγ), CCAAT/enhancer binding protein α, β (C/EBPα, C/EBPβ), insulin-like growth factor I (IGF-l), fatty acids and glucocorticoids. Besides, a range of extracellular signals are also been reported to be engaged in adipogenesis, such as Wingless and INT-1 proteins (Wnts), bone morphogenesis proteins (BMPs), transforming growth factor β (TGFβ) and Hedgehog ligands [2,3]. Hh signaling has been widely reported to negatively regulate the development of adipose tissue [4], however, the role of certain members of this pathway in adipogenesis still remains unclear.
Hhip is a type I membrane glycoprotein consisting of four major domains: the N-terminal hydrophobic stretch, the EGF-like domains, the C-terminal hydrophobic domain and four potential N-linked glycosylation sites [5]. Identified as a potential antagonist of all three mammalian Hh proteins, Sonic (Shh), Indian (Ihh), and Desert (Dhh), Hhip could compete with Patched1 for binding to all three Hh ligands [[5], [6], [7], [8], [9]]. Hhip has been found widely expressed in various cell types, especially in vascular endothelial cells [10]. The functions of Hhip are involved in many biological processes including muscle development [11], pathologic angiogenesis [12], nephropathy development and so on [13]. In a previous meta-analysis, Hhip was identified as closely related to the risk of BMI and type 2 diabetes, which provide robust evidence to implicate it could respond to obesity. Besides, a recent study revealed that Hhip expression was enhanced in glomerular endothelial cells of adult T2D db/db mouse kidneys [13], which further verify Hhip may participate in adipogenesis.