Given an interior operator on the power set of a set X, i.e. a map [itex]\phi[/itex] such that, for all subsets [itex]A,B[/itex] of [itex]X[/itex],(adsbygoogle = window.adsbygoogle || []).push({});

[tex](IO 1)\enspace \phi X = X;[/tex]

[tex](IO 2)\enspace \phi A \subseteq A;[/tex]

[tex](IO 3)\enspace \phi^2A = \phi A;[/tex]

[tex](IO 4)\enspace \phi(A \cap B) = \phi A \cap \phi B,[/tex]

I'm trying to show that the set

[tex]\tau = \left \{ U \in 2^X | \phi U = U \right \}[/tex]

is a topology for [itex]X[/itex]. I've shown everything except that [itex]\tau[/itex] is closed under arbitrary unions. By (IO 2),

[tex]\phi\left ( \bigcup_{\lambda \in \Lambda}U_\lambda \right ) \subseteq \bigcup_{\lambda \in \Lambda}U_\lambda = \bigcup_{\lambda \in \Lambda}\phi U_\lambda.[/tex]

So all that remains is to show that

[tex]\bigcup_{\lambda \in \Lambda}U_\lambda = \bigcup_{\lambda \in \Lambda}\phi U_\lambda \subseteq \phi\left ( \bigcup_{\lambda \in \Lambda}U_\lambda \right ).[/tex]

Any hints? I've used all of the axiom so far except (IO 3), so I'm guessing this must be involved somehow ... I've also done the corresponding exercise for a closure operator and got stuck on at the same point.

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# Topology generated by interior operator

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