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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:
Is the hint misleading me?
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.Quote:
Originally Posted by magus
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.
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.Quote:
Originally Posted by magus