Is the Countable Complement Topology a Valid Topological Space?

[beetle2]

Homework Statement

show that T:=(A subset X |A = 0 or X\A is finite) is a topology on X,

Homework Equations

We need to show 3 conditions.

1: X,0 are in T
2: The union of infinite open set are in T
3: The finite intersections of open sets are open.

The Attempt at a Solution

We see that A \subset X is open in (T1 space) asX\A is finite

To show condition 1
if A = 0 the empty set it is in T
and A\X = X than it is in T.


To show 2

let A \subset X open in T1 as X\A is finite

Let \alpha \in I be an indexing set, A_\alpha \in T so that A \subset X be open as X\A is finite.

Than the \cup_{\alpha \in I} X\A_\alpha = \cap _{\alpha \in I} (X\A_\alpha)

Either each of the sets ( X\A_\alpha) = X , in which case the intersection is all of X, or at least one of them is finite , in which case the intersection is a subset of a finite set and hence finite.

To show 3

Let A_1,A_2,A_3...A_n \subset Xbe open as X\A is finite or all of X.

To show that \cap A_{n} \in Twe must show that \cap X\A_n is either finite or all of X.

But \cap X\A_{n} = \cup X\A_{n}.

Either this set is a union of finite sets and hence finite, or for some X\A_{i} i \in I = Xand the union is all of X.


Thus (A,T) is a topological space.

 

[VeeEight]

You have the right idea for all of the conditions. Applying the set theory properties was the key.

Have you heard of the countable complement topology?