let be $m$ a function "$m: X \to \mathcal{P}(\mathcal{P}(X))$" (and I denote: $m(x) := m_x, \forall x \in X $), $m$ is topology on $X$ if:
2)$ \forall x \in X (\forall t \in m_x(x \in t)) $
3)$ \forall x \in X( X \in m_x) $
4)$ \forall x \in X( \forall t,u \in m_x(t \cap u \in m_x))$
5)$ \forall x \in X (\forall t \in m_x( \exists r \in m_x(\forall y \in r(t \in m_y))))$
6)$ \forall x \in X (\forall t \in m_x( \forall r \subseteq X( t \subseteq r \to r \in m_x)))$
is correct? Thanks in advance!