**Q.** “Do these two sets partitioning a boundary have a common name?” 

**A.** Yes. They are the *rims* of $A$ and its complement.

**Q.** “Who studied them?”

**A.** Kar-Ping Shum

**References**

1. [2017 Shum](http://www.mathtransit.com/cornucopia/2017_shum_a.php) 
2. [1996 Shum](http://www.mathtransit.com/cornucopia/1996_shum_a.php)
3. [1993 Shum](http://www.mathtransit.com/cornucopia/1993_shum_a.php)
4. [1975 Shum and Yip](http://www.mathtransit.com/cornucopia/1975_sy_a.php)

-------------added 7 Oct 2017-----------------

Should have also mentioned: early authors including Kuratowski used the term *border* to describe this operation. It seems to have faded away at some point (probably before Shum and Yip's 1975 paper). I prefer *rim,* because in English, the words “border” and “boundary” generally apply to multiple objects at once (the ones being separated by the border or boundary), whereas something's “rim” applies to itself only.