Geodesic
The sup = max problem for the extent of generalized metric spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 2, pp. 245-257.

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It looks not useful to study the sup = max problem for extent, because there are simple examples refuting the condition. On the other hand, the sup = max problem for Lindelöf degree does not occur at a glance, because Lindelöf degree is usually defined by not supremum but minimum. Nevertheless, in this paper, we discuss the sup = max problem for the extent of generalized metric spaces by combining the sup = max problem for the Lindelöf degree of these spaces.
Classification : 03E10, 54A25, 54D20, 54E18
Keywords: extent; Lindelöf degree; $\Sigma$-space; strict $p$-space; semi-stratifiable 
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Hirata, Yasushi; Yajima, Yukinobu. The sup = max problem for the extent  of generalized metric spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 2, pp. 245-257. http://geodesic.mathdoc.fr/item/CMUC_2013__54_2_a9/