§ 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.

• Pete Clark's notes on point set topology
• Tanuj Gupta: Tychonoff theorem