A system of shrinking neighborhoods
No. 105, 1973
(Mathematical analysis, topology)
India ink and color pencil on paper, 30.5x42.5 cm.
Beneath a hazy sun lies a puddle of tar, sprawled out on the desert floor, like the descendant of a mighty river gone dry. A bar rests in mid- stream. The heat beats down and all is still. Observe that many mathematical constructions use pairs of open sets, such as sets A and B, where B is contained in A along with its own closure. In the construction of a partition of unity, for example, corresponding to a certain cover of a space with a system of open sets, we often see such pairs of sets. This image represents an open set and a contraction of it. The idea is that, given some open cover of a compact topological space, one can shrink slightly each open set in this cover so that the resulting set lies entirely in the original, while the new shrunken sets form an open cover of the original space. Of course, by expanding or contracting such open sets, they can change significantly.
