## § Nets from Munkres (TODO)

#### § Directed set

A direct set is a partial order $J$ which has "weak joins".
That is, for every $a, b \in J$, there exists a $u \in J$ such that $a \leq u$ and $b \leq u$.
It's not a join since we don't need $u$ to be UNIQUE.
#### § Cofinal subset

A subset $K$ of a partial order $J$ is said to be cofinal if loosely, $\forall J \leq \exists K$.
That is, for all $j \in J$, there is a $k \in K$ such that $j \leq k$. So intuitively, $K$
is some sort of portion of $J$ that leaves out a finite part of the bottom of $J$.
#### § Cofinal subset is directed

#### § Nets

Let $X$ be a topological space. a net is a function $f$ from a directed set $J$ into $X$.
We usually write this as $(x_j)$.
#### § Net eventually in a subset $A$

A net $(x_j)$ is eventually in a subset $A$ if there exists an $i \in I$ such that for all
$j \geq i$, $x_j \in A$. This is an $\exists \forall$ formula.
#### § Net cofinally/frequently in a subset $A$

The net $x[:]$ is cofinally in a subset $A$ if the set $\{ i \in I : x_i \in A \}$
is cofinal in $I$. This means that for all $j \in J$, there exists a $k$ such that
$j \leq k$ and $x_k \in A$. This is a $\forall \exists$ formula. So intuitively,
the net could "flirt" with the set $A$, by exiting and entering with elements in $A$.
#### § Eventually in $A$ is stronger than frequently in $A$

Let $x[:]$ be a net that is eventually in $A$. We will show that such a net is also
frequently in $A$. As the net is eventually in $A$, there exists an $e \in I$ (for eventually), such that
for all $i$, $e \leq i \implies x[i] \in A$. Now, given an index $f$ (for frequently),
we must establish an index which $u$ such that $f \leq u \land x[u] \in A$.
Pick $u$ as the upper bound of $e$ and $f$ which exists as the set $I$ is directed.
Hence, $e \leq u \land f \leq u$. We have that $e \leq u \implies x[u] \in A$.
Thus we have an index $u$ such that $f \leq u \land x[u] \in A$.
#### § Not Cofinal/frequently in $A$ iff eventually in $X - A$

#### § Not Frequently in $A$ implies eventually in $X - A$

Let the net be $x[:]$ with index set $I$. Since we are not frequently in $A$,
this means that there is an index $f$ at which we are no longer frequent.
That is, that there does not exist elements
$u$ such that $f \leq u \land x[u] \in A$. This means that for all elements
$u$ such that $f \leq u$, we have $x[u] \not \in A$, or $x[u] \in X - A$.
Hence, we can choose $f$ as the "eventual index", since all elements above $f$
are not in $A$.
#### § Eventually in $X - A$ implies not frequently in $A$

Let the net be $x[:]$ with index set $I$.
Since the net is eventually in $X - A$, this means that
there is an index $e$ (for eventually) such that for all $i$ such that $e \leq i$ we have
$x[i] \in X - A$, or $x[i] \not \in A$. Thus, if we pick $e$ as the
frequent index, we can have no index $u$ such that $e \leq u \land x[u] \in A$,
since all indexes above $e$ are not in $A$.
#### § Convergence of a net

We say a net $(x_j)$ converges to a limit $l\in X$, written as $(x_j) \rightarrow l$ iff
for each neighbourhood $U$ of $l$, there is a lower bound $j_U \in J$ such that for all $k$,
$j_U \leq k \implies x_k \in U$. That is, the image of the net after $j_U$ lies in $U$. In other words,
the net $(x_j)$ is eventually in every neighbourhood of $l$. This is a $\forall \exists \forall$
formula (for all nbhd, exists cuttoff, for all terms above cutoff, we are in the nbhd)
#### § Limit point of a net

We say that a point $l$ is a *limit point* of a net if $x$ is
cofinally/frequently in every neighbourhood of $A$. That is,
for all neighbourhoods $U$ of $A$, for all indexes $j \in J$, there
exists an index $k[U, j]$ such that $j \leq k[U, j] \land x[j] \in U$.
This ia $\forall \forall \exists$ formula (for all nbhd, for all
indexes, there exists a higher index that is in the nbhd).
#### § Converge of a net with net as $\mathbb N$

#### § Product of nets

#### § Convergenge of product nets iff component nets converge

#### § Nets in Hausdorff spaces converge to at most one point

#### § Point $p$ is in closure of $A$ iff net in $A$ converges to point $p$

#### § Function is continuous iff it preserves convergence of nets

#### § Subnets

#### § Subnets of a net converge

#### § Accumulation point of a net

#### § Subnets converge iff point is accumulation point

#### § Compact implies every net has convergent subnet

#### § Compact implies every net has convergent subnet

#### § Universal Net

#### § Every net has universal subnet

#### § Universal net converges in compact space

#### § pushforward of universal net is universal

#### § Tychonoff's theorem

Let $\{ X_\alpha : \alpha \in \Lambda \}$ is a collection of compact
topological spaces. Let $X \equiv \prod_{\alpha \in \Lambda} X_\alpha$ be the
product space. Let $\Phi: D \rightarrow X$ be a universal net for $X$.
For each $\lambda \in \Lambda$, the push forward net $\pi_\lambda \circ \Phi: D \rightarrow X_\lambda$
is a universal net. Thus, it converges to some $x_\lambda \in X_\lambda$.
Since products of nets converge iff their components converge, and here all the components
converge, the original net also converges in $X$. But this means that $X$ is compact
as the universal net converges.