Why extremal regions are interesting?

In the article of Maximally Stable Extremal Region (MSER) [1], why so called "extremal region" are interesting?
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Extremal regions are interesting because the set of extremal regions are closed [2] under

1. Continuous (and thus projective )  transformation of image coordinates
2. Monotonic transformation of image intensities

But what does it mean? What does a close set mean in this case? 
Well, it means any pixel outside the extremal region may not be described by projective transformation (due to viewpoint change). However, pixels inside the extremal regions can be described by projective tranformation. Therefore, Since these extremal regions are affinely covariant, they are more likely to be matched in other views. The whole idea is to find some image regions which are robust to affine covariant transformation. And these extremal regions does that job!

[1] Matas, J.; Chum, O.; Urban, M. & Pajdla, T. Robust wide-baseline stereo from maximally stable extremal regions Image and Vision Computing, Elsevier, 2004, 22, 761-767


[2] Definition of closed set from wiki is "A closed set contains its own boundary. In other words, if you are "outside" a closed set, you may move a small amount in any direction and still stay outside the set."