note-math

net_(tag)

$B$ a net of $𝐴$ := $B \subset Subset (𝐴)$ (a collection of subset of $𝐴$ ) with property

$\forall 𝐵 \in B, 𝐵 \neq \emptyset$
$\forall 𝐵_{1}, \dots, 𝐵_{𝑛} \in B, \exists 𝐵 \in B, 𝐵 \subset 𝐵_{1} \cap \dots \cap 𝐵_{𝑛}$
(Meaning: Non-empty before converging to the limit. If the number of $B$ elements is infinite, then the finite intersection controls the direction of convergence, although possibly $\emptyset = ⋂ B$ . If $B = {𝐵_{1}, \dots, 𝐵_{𝑛}}$ has a finite number of elements, then $\emptyset \neq 𝐵_{1} \cap \dots \cap 𝐵_{𝑛} \in B$ )

Example $𝑋 = ℕ$ , net $B = {𝑛, 𝑛 + 1, \dots}_{𝑛 = 0 . . \infty}$

Point net or net containing point $𝑥 \in 𝑋$ $B (𝑥) ≔ \forall 𝐵 \in B (𝑥), 𝑥 \in 𝐵$

Net $B$ is finer than $B^{'}$ := $\forall 𝐵^{'} \in B^{'}, \exists 𝐵 \in B, 𝐵 \subset 𝐵^{'}$
And this implies $⋂ B \subset ⋂ B^{'}$

net-same-limit_(tag) $B, B^{'}$ have the same limit := are mutually finer than each other

Not all nets with the same limit are useful. Set theoretically, $\cup$ can be used to construct new nets with the same limit, but there is a lot of redundancy

Any net can be supplemented with all finite intersections ${𝐵_{1} \cap \dots \cap 𝐵_{𝑛} : 𝑛 \in ℕ, 𝐵_{1}, \dots, 𝐵_{𝑛} \in B}$ to become a new net, while maintaining the same limit

hom-limit_(tag) Limit homomorphism between nets :=

$𝑓 : 𝑋 \to 𝑌$ . Net $𝑓 (B_{𝑋})$ is finer than $B_{𝑌}$

\begin{matrix} \forall 𝐵_{𝑌} \in B_{𝑌} \\ \exists 𝐵_{𝑋} \in B_{𝑋} \\ 𝑓 (𝐵_{𝑋}) \subset 𝐵_{𝑌} or 𝐵_{𝑋} \subset 𝑓^{- 1} (𝐵_{𝑌}) \end{matrix}

$\subset$ Describing it with points is $\forall 𝑎 \in 𝐵_{𝑋}, 𝑓 (𝑎) \in 𝐵_{𝑌}$

Example

Sequence $ℕ \to ℝ$ converges $lim_{𝑛 \to \infty} 𝑎_{𝑛} = 𝑎$

Using all intervals containing $𝑎$ , $B_{ℝ} = I (𝑎)$ and $B_{ℕ} = {𝑛, 𝑛 + 1, \dots}_{𝑛 = 0 . . \infty}$ , then $ℕ \to ℝ$ is a limit homomorphism

\begin{matrix} \forall 𝐼 (𝑎) \in I (𝑎) \\ \exists {𝑛, 𝑛 + 1, \dots} \in B_{ℕ} \\ {𝑎_{𝑛}, 𝑎_{𝑛 + 1}, \dots} \subset 𝐼 (𝑎) \end{matrix}

$ℝ \to ℝ$ Function limit

let $𝑓 : ℝ \to ℝ, 𝑓 (𝑥) = 𝑦$ , from $I (𝑥)$ to $I (𝑦)$

\begin{matrix} \forall 𝐼_{𝑦} \in I (𝑦) \\ \exists 𝐼_{𝑥} \in I (𝑥) \\ 𝑓 (𝐼_{𝑥}) \subset 𝐼_{𝑦} \end{matrix}