$N(X)$ denotes collection of all nets of $X$. $T:N(X)\to \mathcal{P}(X)$ be a map such that $T\{x_\delta\}=T\{y_\gamma\}$ if one is subnet of other.

Question

Let $X$ be a non-empty set. $N(X)$ denotes the collection of all nets on $X$ and $\mathcal{P}(X)$ denotes the collection of all subsets of $X$. Let $T:N(X)\to \mathcal{P}(X)$ be a map satisfying the condition $T\{x_\lambda\}_{\Lambda}=T\{y_\gamma\}_{\Gamma}$ if one of $\{x_\lambda\}_{\Lambda}$ and $\{y_\gamma\}_{\Gamma}$ is a subnet of the other.
Define $\tau=\{A\subset X|\ T\{x_\lambda\}_{\Lambda}\subset A^c\text{ for every net }\{x_\lambda\}_{\Lambda}\subset A^c\}$

We have to prove that $\tau$ is a topology on $X$. I have tried the problem in the following manner-

1. $\emptyset\in \tau$ since given any net $\{x_\lambda\}_{\Lambda}\subset (\emptyset)^c=X$, we always have $T\{x_\lambda\}_{\Lambda}\subset (\emptyset)^c=X$ . Again, $X\in \tau$ since $X^c=\emptyset$
2. Let $\{A_i|\ i\in I\}\subset \tau$. Then let $\{x_\lambda\}_{\Lambda}\subset \left(\bigcup A_i\right)^c=\bigcap A_i^c\implies \{x_\lambda\}_{\Lambda}\subset A_i^c\ \forall i\implies T\{x_\lambda\}_{\Lambda}\subset A_i^c\ \forall i\implies T\{x_\lambda\}_{\Lambda}\subset \bigcap A_i^c=\left(\bigcup A_i\right)^c$. Hence, $\bigcup A_i\in \tau$.
3. Let $A,B\in \tau$ and $\{x_\lambda\}_{\Lambda}\in (A\cap B)^c=A^c\cup B^c$. Now we have to prove $T\{x_\lambda\}_{\Lambda}\subset A^c\cup B^c$.
   Now I define $\Gamma=\{\lambda\in\Lambda|\ x_\lambda\in A^c\}, \Sigma=\{\lambda\in\Lambda|\ x_\lambda\in B^c\}$. Now if I can prove at least one of $\Lambda_1,\Lambda_2$ is directed set then I will construct a subnet of $\{x_\lambda\}_{\Lambda}$, and use the property of $T$ to achieve our goal. But I'm to prove that at least one of $\Gamma,\Sigma$ is directed set, although I have shown that both $\Gamma, \Sigma$ satisfies reflexivity and transitivity condition. But we have to show the third condition which is $$\text{If }\lambda_1,\lambda_2\in \Gamma\text{ (or }\Sigma\text{)}, \exists \lambda_3\in \Gamma\text{ (or }\Sigma\text{) such that } \lambda_3\ge\lambda_1,\lambda_2$$

Can anyone help me with this regard? Thanks for your help in advance.

Answer 1

$\Gamma \cup \Sigma = \Lambda$ so one of them is cofinal in $\Lambda$ ( if neither, then the directedness gives an immediate contradiction ), so say that $\Gamma$ is cofinal. But then $\Gamma$ defines a subnet of $x$ living on $A^c$ (the cofinality ensures its also directed as a suborder) and so has the same $T$ set as $x$, and as $A\in \tau$ that $T(x) \subseteq A^c$ and we’re done.