Countability Axioms and Nets

Countability Axioms

First Countable

A topological space is first-countable if there is a countable local basis at each point . That is, for each , there is a sequence of open neighbourhoods such that for any open neighbourhood of there is some so that .

e.g.

• Any metric space is first countable. To see this, note that the set of open balls centered at with radius for all natural numbers form a countable local basis at ;
• Cofinite topology on uncountable sets is not first countable;
  

Proposition

If a topological space is first countable, then it has a countable nested local basis
for all .

Proof This is because any countable local basis

Second Countable

A topological space is second-countable if there is a countable basis for .

e.g. with the usual topology is second-countable, since the collection of all open balls with rational radius and rational center forms a countable basis.

Proposition

If is second countable, then

1. is first countable;
2. is separable;
3. is Lindelöf.
   

Proof

Nets

In the topology scenario, a net is a generalization of a sequence:

Net

Suppose is a topological space, then a net in is a map , where is a directed set. We often write it as .

Convergence of Nets

Let be a topological space and be a net in . Then we say that converges to if for every neighbourhood of , there exists such that for all . We write or simply .

Subnet

Let be a net in . A subnet of is a net in such that there exists a map satisfying and for all , there exists such that for all .

We now show that the similar results for sequences also hold for nets.

Proposition

Let be a topological space, and . Then if and only if there exists a net in such that .

Proof Suppose . Then for every neighbourhood of , we have . Let , and define a preorder on by if and only if . Then is a directed set. For each , choose . Then is a net in that converges to . The converse is clear.