Explanation: Fourteen Sets Formed By Closure, Interior and Complement Operations

(related to Chapter: Basic Topological Concepts)

If $A$ is a set, we can create at most fourteen sets by successive application of the closure $A^-$, interior $A^\circ$ and complement $A^C$ operations. These fourteen sets are:

  1. $A$
  2. $A^C$
  3. $A^\circ=A^{- C -}$
  4. $A^-$
  5. $A^{\circ C}$
  6. $A^{C-}$
  7. $A^{\circ-}$
  8. $A^{-\circ}$
  9. $A^{\circ-\circ}$
  10. $A^{\circ C\circ}$
  11. $A^{C\circ-}$
  12. $A^{-\circ -}$
  13. $A^{-\circ C\circ}$
  14. $A^{\circ -\circ C}$

It is left to the reader as an exercise to build these 14 sets from this example subset of real numbers $\mathbb R$.

$$A:=\{\frac 1n\mid n=1,2,3\ldots\}\cup (2,3)\cup (3,4)\cup\left\{4 \frac 12\right\}\cup [6,7] \cup \{x\in \mathbb Q\mid 8\le x < 9\}$$

The example set is chosen in such a way that indeed all 14 sets are different. $(a,b)$ and $[a,b]$ denote open and closed real intervals.

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  1. Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984
  2. Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
  3. Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition