# isomorphism filter and nets

Given a filterbasis, by picking up an point out of every element of this filterbasis, and considering the ordering on the elements of the filterbasis $$u\leq v$$ iff $$v \subset u$$ (subquestion why is it not the other way like subset v) , we get a net.

Given a net over I, we get a filterbasis by considering $$\{ x_k| k \geq i \}$$,for all $$i \in I$$.

Can we write a isomorphism between the set of all filters and the set of all nets. Or is there a filter not derivable from a net (or vice versa).

here they claim there is a Galois connection, i cant see the pdf an am curious how that actually would be prooven, if there is no isomorphism.

• The excellent PDF by Saitulaa Naranong can still be seen via the WayBack Machine. Note that $\Psi$ and $\Phi$ have been inadvertently interchanged in the displayed line in Definition $\mathbf{10.2}$ at the top of page $11$. The one-sentence paragraph two lines down (‘In other words ...’) is correct. – Brian M. Scott Jun 15 at 16:09
• And yes, there is a natural correspondence between the two; see Lemma $\mathbf{5.3}$. For your subquestion, think about convergence: it’s the tails of the net that have to get close to the limit point, and it’s the smaller nbhds that are close to that point. – Brian M. Scott Jun 15 at 16:15

More precisely, if $$\mathcal{F}$$ is a filterbase on $$X$$, then on the directed set

$$I(\mathcal{F}) = \{(x,F): x \in F \in \mathcal{F}\}; (x_1,F_1) \le (x_2, F_2) \iff F_2 \subseteq F_1$$

we have a net $$\Phi(\mathcal{F}); I(\mathcal{F}) \to X; (x,F) \to x$$. (so we consider all choices for points from $$F$$ simultaneously, so no AC is needed in the definition, it's all canonical).

And indeed we have an inverse by using tails:

If $$f: I \to X$$ is a net defined on some directed set $$I$$, we define

$$\mathcal{F} = \{ \{f(i): i \in I, i \le i_0\} \mid i_0 \in I\}$$

the tail filter $$\Psi(f)$$ defined by $$f$$. And indeed these maps are each other's inverse (see this link (thx to Brian) for complete proofs).

This correspondence is "Galois" in the order reversal properties : a subnet of a net corresponds (under this map) to a larger filterbase, and vice versa, a larger filter(base) makes for a subnet.

• What do you mean whit AC? – user23657 Jun 16 at 2:29
• @user23657 the Axiom of Choice. – Henno Brandsma Jun 16 at 4:11
• See here (wiki) e.g. – Henno Brandsma Jun 16 at 4:49