Directed Sets

Definition 1.7.5   $D$ is a directed set if $D$ has a partial order (transitivity, reflexive, anti-symmetric) such that for all $\alpha, \beta \in D$, there exists $\gamma \in D$ such that $\alpha \leq \gamma$, $\beta \leq \gamma$.

Examples:

1. $\mathbb{N}$
2. Fix $X$ in a topological space $X$; $\{ U$    open $: x \in U \}$ ordered by reverse inclusion.
3. Let $X$ be a set. Let $\mathcal{S}$ be the finite subsets of $X$ ordered by inclusion.
4. Let $[a,b]$ be an interval in $\mathbb{R}$. Take the set of partitions of $[a,b]$ ordered by inclusion.

Definition 1.7.6   A net is a function from a directed set to a topological space $f:D \rightarrow X$. Write $x_\lambda$ for $f(\lambda)$.

Basic Example: Given $x \in X: U \in N(x)$. Let $x_U$ be some point of $U$.

Definition 1.7.7   $x_\lambda$ converges to $x$ if $\forall U \in N(x)$, there exists $\lambda_0 \in D$ such that $\forall \lambda \geq \lambda_0$, $x_\lambda \in U$.

Take a basic example with $X = \mathbb{R}$, $x = 0$, and require $\vert x_U\vert \geq \min \{ 1, \frac12 \sup \vert U\vert \}$.

1. This net is unbounded. For $N \in \mathbb{N}$, take $U = (-1,2N+1)$. Then, $x_U = N + \frac12 > N$.
2. This net converges to 0. Let $U = N(0)$, i.e. an open set containing 0. Let $\lambda_0 = U$. Then if $\lambda \in D$, $\lambda \geq \lambda_0$, then $\lambda$ is an open set contained in $\lambda_0$ with $0 \in \lambda$. Thus, $x_\lambda \in \lambda \subset \lambda_0 = U$. So, $x_U \rightarrow 0$.

Let $A \subset X$.

Proposition 1.7.8   $x \in \bar A$ iff $\exists$ net $x_\lambda$ in $A$ converging to $X$.

Moral:
Almost every result about sequences in a metric space generalizes to nets in a topological space (except for those ones we've already shown).

Counter Example:
Let $X$ be uncountable set with a total order. Then, for all $x \in X$, $\{y \in X : y \leq x \}$ is countable. Let $Y = X \cup \{ \infty \}$ with $x < \infty$ for all $x \in X$.
Facts:

1. $\infty \in \bar X$.
2. No sequence in $X$ converges to $\infty$ (cardinality element).

Monday, 1-30-2006:

We had a result, $x \in \bar A$ iff there is a net in $A$ converging to $X$ and an example to show we can't change net to sequence in a general topological space.

Definition 1.7.9   We say that a topological space satisfies ``the first axiom of countability" if for all $x \in X$, there exists a countable subset of $N(x)$, $(u_n)$, such that $\forall V \in N(x)$, there exists $n \in \mathbb{N}$ such that $U_n \subset V$.

Example: Metric space. Choose $B_\frac1n(x)$, $n \in \mathbb{N}$.

If a topological space satisfies this axiom, then we can change net to sequence in the result. The proof is an exercise.

Continuity:
$f:X \rightarrow Y$ is continuous iff $f^{-1}(U)$ is open in $Y$ for all $U \subset X$ open. $f$ is continuous at $x_0 \in X$ iff for all $V \in N(f(x_))$, there is $U \in N(x_0)$ such that $f(U) \subset V$.

Proposition 1.7.10   $f$ is continuous at $x_0$ iff for all nets $x_\lambda \rightarrow x_0$, $f(x_\lambda) \rightarrow f(x_0)$.

We have two ways to build topological spaces.

1. Initial Topology Let $X$ be a set, and $f:X \rightarrow Y_f$ for $f \in \mathcal{F}$. Each $Y_f$ has a topology $\tau_f$. The smallest topology on $X$ such that each $f$ is continuous has as a subbasis:
   $$\{ f^{-1}(V) : V \in \tau_f, f \in \mathcal{F}\}$$
   Proposition 1.7.11   $x_\lambda \rightarrow x_0$ in this topology iff $f(x_\lambda) \rightarrow f(x_0)$. $g:Z \rightarrow X$ is continuous iff $f \circ g:Z \rightarrow Y_f$ is continuous for all $f \in \mathcal{F}$.
   Examples:
   
   1. Let $X_\lambda:\lambda \in \Lambda$ be a collection of topological spaces. Let
      $$X = \prod_{\lambda \in \Lambda} x_\lambda := \{ f: \lambda \rightarrow \bigcup x_\lambda : f(\lambda) \in X_\lambda \}.$$
      This is the prodoct topology on $X$. This is the initial topology for $\pi_\lambda : X \rightarrow X_\lambda$ given by $\pi_\lambda(f) = f(\lambda)$.
   2. Given a vector space $X$ over $\mathbb{F}$ a collection of functionals $F$, the $\sigma(X,F)$ topology is exactly the initial topology for $\{ f : X \rightarrow \mathbb{F}: f \in \mathbb{F}\}$.
2. Final Topology: Let $Y$ be a set. Let $f: X_f \rightarrow Y$ for $f \in \mathcal{F}$. Let $X_f$ have a topology $\tau_f$. The final topology on $Y$ is the largest topology so that each $f \in \mathcal{F}$ is continuous; i.e.,
   $$\{ A \subset Y : f^{-1}(A) \in \tau_f$$    for all $$f \}$$
   Proposition 1.7.12   $g:Y \rightarrow Z$ is continuous iff $g\circ f : X_f \rightarrow Z$ is continuous for all $f \in \mathcal{F}$.
   The quotient topology on $X /$ is the final topology for $\{ Q \}$.

Asides:
Points in $X /$ are closed iff each equivalent class is closed. $Q$ is an open map iff for all open sets $A \subset X$, $\{ x \in A : x  m$    for some $a \in A \}$ is open.

Fact: If $X$ LCS, $M \leq X$, and $A \subset X$, open,

$$\{ x \in X : x - a \in M$$    for some $$a \in A \}.$$

$$= \{ x \in X : x \in A+m$$    for some $$m\in M \}.$$

$$= \bigcup_{m \in M} A+ m$$

is open.