Nets - preoredered sets or posets?

Solution 1:

In Hausdorff spaces, we probably lose little either way. In the "partial order" setting we require the axiom of choice more often. (Did you see "choose" in your write-up? Now do the same thing WITHOUT the axiom of choice...)

There is more than one paper where the author claims something doesn't work for nets (in order to justify his/her use of filters) by incorrectly requiring the partial-order version of directed set.

One use of nets (indeed, the original use by Moore and Smith) is to explain the convergence used in the definition of the Riemann integral, where you have a Riemann sum for each tagged partition of an interval, and you direct them by refinement, allowing any choice of tags. Preordered only!

added
This came up again, so let me amplify my remark about non-Hausdorff spaces, and avoiding the axiom of choice (AC).

Let $(X, \mathscr X)$ and$(Y, \mathscr Y)$ be topological spaces, let $f : X \to Y$ be a function, let $a \in X$. For $a \in X$ let $\mathscr X_a = \{U \in \mathscr X : a \in U\}$ be the set of open neighborhoods of $a$; similarly for $\mathscr Y_b$. We say that $f$ is continuous at $a$ iff:

for every $V \in \mathscr Y_{f(a)}$ there exists $U \in \mathscr X_a$ such that $f(U) \subseteq V$.

But if we like to use nets, we want to say: this is equivalent to

for every net $(p_d)_{d \in D}$ in $X$, if $p_d \to a$ then $f(p_d) \to f(a)$.

To avoid AC we do it like this: Suppose $f : X \to Y$ is discontinuous at $a$. We want to find a net with $p_d \to a$ but $f(p_d) \not\to f(a)$. Easy: Let $$ D = \big\{(U,x) : x \in U, x \in U \in \mathscr X_a\big\} \tag1$$ ordered by $$ (U_1,x_1) \le (U_2, x_2) \quad\Longleftrightarrow\quad U_1 \supseteq U_2. \tag2$$ This is directed because: the intersection of two open sets is open. [We avoided AC here; we did not choose an element of $U$, instead we use all of the elements.]

Define a net $(p_d)_{d \in D}$ by $p_{(U,x)} = x$. Then we can show $x_d \to a$. Indeed, given $U \in \mathscr X_a$, we have $p_d \in U$ for all $d \ge (U,a)$.

Also we claim $f(p_d) \not\to f(a)$. We assumed $f$ is discontinuous at $a$. So there is $V$ with $f(a) \in V \in \mathscr Y_{f(a)}$ but for any $U \in \mathscr X_a$ we have $f(U) \not\subseteq V$. That is: there is $(U,y) \in D$ with $f(p_{(U,y)}) \notin V$.

Remarks. Notice that, in a non-Hausdorff space, it could be that there is a minimal open set with more than one element. The construction above still works. But the directed set ($1$) ordered by ($2$) is not partially ordered. Only quasi-ordered.

The originating material on nets (Birkhoff 1950, then Kelley 1955) used quasi-ordered directed sets, so everything worked. Subsequent readers of their expositions have sometimes not noticed this.