Let $X$ be a set. A property $p$ of $X$ is a function

An element $x\in X$ is said to have or does not have the property $p$ depending on whether $p(x)=\mathit{true}$ or $p(x)=\mathit{false}$. Any property gives rise in a natural way to the set

and the corresponding characteristic function $1_{{X(p)}}$. The identification of $p$ with $X(p)\subseteq X$ enables us to think of a property of $X$ as a 1-ary, or a unary relation on $X$. Therefore, one may treat all these notions equivalently.

Usually, a property $p$ of $X$ can be identified with a so-called propositional function, or a predicate $\varphi(v)$, where $v$ is a variable or a tuple of variables whose values range over $X$. The values of a propositional function is a proposition, which can be interpreted as being either “true” or “false”, so that $X(p)=\{x\mid\varphi(x)\mbox{ is }\mathit{true}\}$.

Below are a few examples:

• Let $X=\mathbb{Z}$. Let $\varphi(v)$ be the propositional function “$v$ is divisible by $3$”. If $p$ is the property identified with $\varphi(v)$, then $X(p)=3\mathbb{Z}$.

• Again, let $X=\mathbb{Z}$. Let $\varphi(v_{1},v_{2}):=$“$v_{1}$ is divisible by $v_{2}$” and $p$ the corresponding property. Then

  $$X(p)=\{(m,n)\mid m=np\mbox{, for some }p\in\mathbb{Z}\},$$

  which is a subset of $X\times X$. So $p$ is a property of $X\times X$.

• The reflexive property of a binary relation on $X$ can be identified with the propositional function $\varphi(V):=``\forall a\in X\mbox{, }(a,a)\in V$”, and therefore

  $$X(p)=\{R\subseteq X\times X\mid\varphi(R)\mbox{ is }\mathit{true}\},$$

  which is a subset of $2^{{X\times X}}$. Thus, $p$ is a property of $2^{{X\times X}}$.

• In point set topology, we often encounter the finite intersection property on a family of subsets of a given set $X$. Let

  $$\varphi(\mathcal{V}):=``\forall n\in\mathbb{N},\forall E_{1}\in\mathcal{V},\ldots,\forall E_{n}\in\mathcal{V},\exists x\in X(x\in E_{1}\cap\cdots\cap E_{n})\mbox{''}$$

  and $p$ the corresponding property, then

  $$X(p)=\{\mathcal{F}\subseteq 2^{X}\mid\varphi(\mathcal{F})\mbox{ is }\mathit{true}\},$$

  which is a subset of $2^{{2^{X}}}$. Thus $p$ is a property of $2^{{2^{X}}}$.