In closure spaces (thus, also in topological spaces), one may define the boundary of a set A as the closure of A minus the interior of A. This set is partitioned into "the closure of A minus A" and "A minus the interior of A", that are equal, respectively, to "the boundary of A intersected with the complement of A" and "the boundary of A intersected with A".


Do these two sets partitioning a boundary have a common name? Who studied them?



Name of the concept "Topological boundary of A intersected with A"
**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.


