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## Homework Statement

Prove that T

_{1}={U subset of X: X\U is finite or is all of X} is a topology.

## Homework Equations

DeMorgan's Laws will be useful.

Empty set is defined as finite, and X is an arbitrary infinite set.

## The Attempt at a Solution

1) X/X = empty set, finite. Thus X is in T

_{1}

2) X/empty set = X, infinite. Thus Empty Set is in T

_{1}

3) Let {U

_{a}:a is in A} be a collection of sets in T

_{1}

Spse U

_{a}≠ X.

X\(indexed unionU

_{a}) = indexed intersection(X\U

_{a})

This is finite since each X\U

_{a}is finite or is all of X. (element of T

_{1})

Spse U

_{a}= empty set for all a, then the indexed intersection(X\U

_{a}) = X (element of T

_{1})

Spse U

_{a-bar}≠ empty set for some a, then the indexed intersection(X\U

_{a}) is a subset of X\U

_{a-bar}(element of T

_{1}).

4) Let {U

_{a}:a is in A} be a collection of sets in T

_{1}

Spse U

_{a}≠ empty set.

X\(indexed intersectionU

_{a}) = indexed union(X\U

_{a}),

Since the number of intersection must be finite and X\U

_{a}is finite for all a,

Then the union of finitely many finite sets is finite. (element of T

_{1})

If U

_{a}= empty set for some a,

Then the indexed union (X\U

_{a}) is either finite or all of X. (element of T

_{1})