User: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 $$\exists !$$. $$\begin{align} \forall w_1,\ldots,w_n \, \forall A \, ( [ \forall x \in A. &\, ! y. \, \phi(x, y, w_1, \ldots, w_n, A) ]\ \Longrightarrow\ \exists B \, \forall y \, [y \in B \Leftrightarrow \exists x \in A \, \phi(x, y, w_1, \ldots, w_n, A) ] ) \end{align}$$

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 $$\mbox{Top}$$.