Is the definition of a topology on a set an impredicative one?

Question

Given that a topology on a set X is defined as a subset of the powerset of X that satisfy certain conditions and the powerset of X itself is a topology over X, I conclude, that the definition is impredicative. Is this line of reasoning correct?

Answer 1

The notion of impredicativity is context-dependent and not always clear, but in this case, I don't think you can argue that because the powerset $\Bbb{P}(X)$ of a set $X$ gives a topology on $X$ (the indiscrete topology), then the definition of a topology as a subset ${\cal T}$ of $\Bbb{P}(X)$ satisfying certain conditions is impredicative. That would be analogous to arguing that because the number $2$ is even, then the definition of an even number as a number divisible by $2$ is impredicative.

To relate this to your quotation from the Encyclopedia of Mathematics, note that $2$ in the definition of even-ness and $\Bbb{P}$ in the definition of topology are not bound variables: $2$ is a constant symbol and $\Bbb{P}$ is a function symbol. The "object to be defined" in the definition of a topology is the class of all topologies. The definition does not involve quantification over this class.

Answer 2

Following up on Rob Arthan's answer I attempt a more detailed one:

To begin with, Carnap wrote in the Logical Syntax of Language: “A thing is called impredicative (in the material mode of speech) when it is defined (or can only be defined) with the help of a totality to which it itself belongs. This means (translated into the formal mode of speech) that a defined symbol $\mathfrak{a}_1$ is called impredicative when an unrestricted operator with a variable to whose range of values $\mathfrak{a}_1$ belongs, occurs in its chain of definitions” (p.163)

The term “operator” above, signifies a quantifier. In the case of sets the aforementioned definition should be understood as follows: a set $X$ defined as follows,$(\forall x)(x\in X\leftrightarrow\ A\left(x\right))$ for some predicate $A$, is called impredicable if for the formulation of $A$ are used quantifiers that bound variables with range of values which includes the definiendum $X$.

Let us now try to define the set of topologies $\mathcal{T}$ on a set $X$:

$(\forall x)[x\in\mathcal{T}\leftrightarrow x\subseteq\mathbb{P}(X)\land\ \emptyset\in x\land\ Χ∈x∧(∀y)(∀z)((y∈x∧z∈x)→y∩z∈x)∧(∀w)(∃φ)(∀t)((t∈w↔(t∈x∧(∃i)(i∈\mathbb{N}∧φ(i,t)))→∪w∈x)]$

where $\mathbb{P}(X)$ the power set of $X$, $\mathbb{N}$ the set of natural mumbers and $\varphi(s,t)$ any function that defines a countable family of subsets of $X$.

It can be easily verified that the definition is not impredicative since $\mathcal{T}$ is not the value of any bound variable in the definiens to the right of the first occurrence of the equivalence.