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Homework Statement
Verify that taking [tex]\mathbb{R}[/tex], the empty set and finite sets to be closed gives a topology.
Homework Equations
The Attempt at a Solution
Clearly the empty set is finite as it has 0 elemnts, and so is closed.
If [tex] X_i [/tex], for i= {1,...,n}, are finite sets then clearly the union of finitely many finite sets is again finite and so is closed.
Let [tex]|X_i| = m_i [/tex], where [tex]m_i[/tex] is the number of elements in [tex]X_i[/tex], then [tex]|\bigcap X_i |[/tex] is at most max{[tex]m_i[/tex]} or at least 0 if the intersection is empty. Either way it is again finite and so is defined as closed.
I think I have shown the empty set, arb unions, arb intersections are closed in this topology, but I can't see how [tex] \mathbb{R}[/tex] could be included...
Thanks