# How to show that any Kuratowski interior induces a Kuratowski closure?

At the 2th chapter of the "Elementos de Topología General" by Fidel Casarrubias Segura and Ángel Tamariz Mascarúa it is written that the interior operator $$\eta$$ induces a topology and to prove this it say that defining the operator $$\kappa:\mathcal{P}(X)\owns{E}\rightarrow{X\setminus{\eta(E)}}\in\mathcal{P}(X)$$ it is possible to demonstrate that $$\kappa$$ is a closure operator and so $$\eta$$ induce a topology on X that is the same of closure operator $$\kappa$$.

Well since $$\eta$$ is an interior operetor I know that:

1. $$\eta(X)=X$$;
2. $$\eta(E)\subseteq{E}$$ for any $$E\subseteq{X}$$;
3. $$\eta(\eta(E))=\eta(E)$$ for any $$E\subseteq{X}$$;
4. $$\eta(A\cap{B})=\eta(A)\cap{\eta(B)}$$ for any $$A,B\subseteq{X}$$.

So to demonstrate that $$\kappa$$ is a closure operetor I must demonstrate that:

1. $$\kappa(\varnothing)=\varnothing$$;
2. $$E\subseteq\kappa(E)$$ for any $${E}\subseteq{X}$$;
3. $$\kappa(\kappa(E))=\kappa(E)$$ for any $${E}\subseteq{X}$$;
4. $$\kappa(A\cup{B})=\kappa(A)\cup\kappa(B)$$ for any $$A,B\subseteq{X}$$.

Howewer first it seems to me that $$\kappa(\varnothing)=X\setminus\eta(\varnothing)=X\setminus\varnothing=X\neq\varnothing$$ and then I can't prove the other points so I only see that

1. $$\eta(E)\subseteq{E}\Rightarrow{X\setminus{E}}\subseteq{X\setminus\eta(E)}=\kappa(E)$$;
2. $$\kappa(\kappa(E))=\kappa(X\setminus\eta(E))=X\setminus\eta(X\setminus\eta(E))$$;
3. $$\kappa(A\cup{B})=X\setminus\eta(A\cup{B})$$.

Could someone help me?

Furthermore how to demonstrate that $$int(E)=\eta(E)$$?

• The correct formula is $\kappa(E)=X\setminus \eta(X \setminus E)$ Jan 30 '20 at 12:18

If $$\eta$$ is an interior operator, you have to define $$\kappa(E)= X\setminus \eta(X\setminus E)$$ and then e.g.
$$\kappa(E) = X\setminus \eta(X\setminus E) \supseteq X \setminus( X \setminus E) = E$$ etc. Use that $$\setminus$$ reverses inclusions and changes $$\cap$$ into $$\cup$$ and vice versa by de Morgan's laws.