# Discrete topology on an uncountable set

• Apr 9th 2011, 02:22 PM
magus
Discrete topology on an uncountable set
Prove that the discrete topology on an uncountable set does not satisfy the second axiom of countability.

The hint says: It is not enough to prove that one basis is uncountable but every basis is uncountable.

I can however only think of one possible basis and it's uncountable: $\displaystyle \{\{x\}|x\in X\}$

• Apr 9th 2011, 02:31 PM
magus
As soon as I typed it out I thought of other baseses (what's the plural for this)?

Every basis though has to contain that basis so it will be uncountable. Is that right?
• Apr 9th 2011, 11:02 PM
Drexel28
Quote:

Originally Posted by magus
As soon as I typed it out I thought of other baseses (what's the plural for this)?

Every basis though has to contain that basis so it will be uncountable. Is that right?

Bases.

The easy proof is that if it were second countable then it would be separable and so have a countable dense subset but (evidently?) the only dense subset is the full space which is uncountable.
• Apr 10th 2011, 06:02 AM
magus
I'm not sure what separable means and I don't think I can use that.
• Apr 10th 2011, 09:54 AM
Drexel28
Quote:

Originally Posted by magus
I'm not sure what separable means and I don't think I can use that.

It means countable dense set and I bet you can.