1. ## Countable Topology

X is a set and $\displaystyle \tau_c$ is a collection of all the subsets U of X such that X-U either is countable or X.

$\displaystyle X-\bigcup_{\alpha\in A}U_{\alpha}=\bigcap_{\alpha\in A}(X-U_{\alpha})$. (DeMorgan's Law--I understand this)

Now, it says the complement of $\displaystyle \bigcup_{\alpha\in A}U_{\alpha}$ is a subset of countable sets.

I don't get this. Can someone explain what is going on?

Thanks.

2. ## Re: Countable Topology

Originally Posted by dwsmith
X is a set and $\displaystyle \tau_c$ is a collection of all the subsets U of X such that X-U either is countable or X.
$\displaystyle X-\bigcup_{\alpha\in A}U_{\alpha}=\bigcap_{\alpha\in A}(X-U_{\alpha})$. (DeMorgan's Law--I understand this)
Now, it says the complement of $\displaystyle \bigcup_{\alpha\in A}U_{\alpha}$ is a subset of countable sets.
Disclaimer: I am not at all sure that I follow this question.
$\displaystyle \left( {\bigcup {U_\alpha } } \right)^c = \bigcap {\left( {U_\alpha } \right)^c = \bigcap {\left( {X\backslash U_\alpha } \right)} }$

3. ## Re: Countable Topology

How do we know or how can it be shown that the intersection is countable?

4. ## Re: Countable Complement Topology

Originally Posted by dwsmith
X is a set and $\displaystyle \tau_c$ is a collection of all the subsets U of X such that X-U either is countable or X.

$\displaystyle X-\bigcup_{\alpha\in A}U_{\alpha}=\bigcap_{\alpha\in A}(X-U_{\alpha})$. (DeMorgan's Law--I understand this)

Now, it says the complement of $\displaystyle \bigcup_{\alpha\in A}U_{\alpha}$ is a subset of countable sets.

I don't get this. Can someone explain what is going on?

Thanks.
It looks like you're working with the Countable Complement Topology.

Is this part of an exercise in which you show that the collection $\displaystyle \tau_c$ is a topology on X by showing that an arbitrary union of sets that are in $\displaystyle \tau_c$, is itself in $\displaystyle \tau_c$?

5. ## Re: Countable Topology

Originally Posted by dwsmith
How do we know or how can it be shown that the intersection is countable?
Any subset of a countable set is countable.
An intersection of a collection of sets is a subset of each set in the collection.

6. ## Re: Countable Topology

Originally Posted by Plato
Any subset of a countable set is countable.
An intersection of a collection of sets is a subset of each set in the collection.
Why is the set countable instead of not countable (that is what I don't understand)?

7. ## Re: Countable Topology

Originally Posted by dwsmith
Why is the set countable instead of not countable (that is what I don't understand)?
Look carefully at the definition.
For each $\displaystyle \alpha$ the set $\displaystyle X\setminus U_{\alpha}$ is either $\displaystyle X$ or is countable.
Thus each set in that intersection is either $\displaystyle X$ or is countable.

8. ## Re: Countable Topology

Originally Posted by Plato
Look carefully at the definition.
For each $\displaystyle \alpha$ the set $\displaystyle X\setminus U_{\alpha}$ is either $\displaystyle X$ or is countable.
Thus each set in that intersection is either $\displaystyle X$ or is countable.
Ok thanks. I wasn't sure I could assume that from the definition. I thought I would have to show it was countable and fit into the definition to be true.