Cofinite Topology: Borel Algebra

Question

Given the cofinite topology: $$\mathcal{T}:=\{U\subseteq\Omega:\#U^c<\infty\}$$ and generate its Borel algebra: $$\sigma(\mathcal{T})=\{E\subseteq\Omega:\#E\leq\aleph_0\lor\#E^c\leq\aleph_0\}$$ Why is this its Borel algebra?

Intuitively, this makes sense but rigorously?

Ah you mean like check that this is a sigma algebra and check that any sigma algebra containing the topology must contain that sigma algebra concluding it is the one genereated by the topology, right? — freishahiri, Oct 23 '14 at 18:37
Can you put it as answer, please? (So I can check it off.) — freishahiri, Oct 24 '14 at 06:47

score 1 · Accepted Answer · answered Oct 27 '14 at 12:59

1

Without proof:

It is a sigma algebra.

It contains the cofinite topology.

Any sigma algebra containing the cofinite topology must in fact contain it.

Thus, it is its Borel algebra by definition.

answered Oct 27 '14 at 12:59

freishahiri

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Cofinite Topology: Borel Algebra

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