Georgydunaev

empty

The exclamation mark($!$), can be also used as a separate quantification symbol, so $(\exists !x.P(x))\leftrightarrow ((\exists x.P(x))\land (!x.P(x)))$, where $(!x.P(x)):=(\forall a\forall b.P(a)\land P(b)\rightarrow a=b)$. E.g. it can be safely used in the replacement axiom, instead of $\exists !$. ${\begin{aligned}\forall w_{1},\ldots ,w_{n}\,\forall A\,([\forall x\in A.&\,!y.\,\phi (x,y,w_{1},\ldots ,w_{n},A)]\ \Longrightarrow \ \exists B\,\forall y\,[y\in B\Leftrightarrow \exists x\in A\,\phi (x,y,w_{1},\ldots ,w_{n},A)])\end{aligned}}$

A topology is exactly an element of the proper class ${\mbox{Top}}$, which is defined the following way: ${\mbox{Top}}=\{{\mathcal {T}}:(\forall A,B\in {\mathcal {T}}.A\cap B\in {\mathcal {T}})\land (\forall U\subseteq {\mathcal {T}}.\bigcup U\in {\mathcal {T}})\}$. The underlaying set $X$ can be unambiguously reconstructed from the topology, because $X=\bigcup {\mathcal {T}}$. Propeties $\emptyset \in {\mathcal {T}}$ and $X\in {\mathcal {T}}$ are simple theorems, which follow from the right part of the conjunction in the definion of ${\mbox{Top}}$.