---
tag: NBYFT0BFKX
kind: statement
role: proposition
name: Visualization of EF-axiom
sources:
  - wikipedia: en:23110101@r1375612711
    locator: Near sets, section Visualization of EF-axiom
    note: adopted
    mention: wikipedia://en/23110101#IYPM
    sections: [Statement]
inventory: [73WJ107JZ3]
---

## Statement

Let the set $X$ be represented by the points inside the rectangular region in Fig. 5. Also, let $A,B$ be any two non-intersection subsets (*i.e.* subsets spatially far from each other) in $X$, as shown in Fig. 5. Let $C^c = X \backslash C$ ([[complement|0RKF1ZT5JJ]] of the set $C$). Then from the EF-axiom, observe the following:

$$\begin{aligned} A &{}\mathrel{\underline{\delta}} B,\\ B &\subset C,\\ D &= C^c,\\ X &= D \cup C,\\ A &\subset D,\ \mbox{hence, we can write}\\ A\ \underline{\delta}\ B\ &\Rightarrow\ A\ \underline{\delta}\ C\ \mbox{and}\ B\ \underline{\delta}\ D,\ \mbox{for some}\ C,D\ \mbox{in}\ X \mbox{ so that } C\cup D = X. \qquad \blacksquare \end{aligned}$$