Can directed set be finite?

Question

Can we have directed sets to be finite sets?

How do we define nets on such sets?

How do we define clustering and convergence of such nets?

Answer 1

Note that in a directed set every finite subset has an upper bound. If the directed set itself is finite, then it must have a maximum (why?). Therefore the limit of such net is the element associated with the maximum index.

Convergence is trivial in that case.

Answer 2

The empty set $\emptyset$ is directed (if empty set is allowed). A non-empty directed set is $(\{1\},1\le1)$. If a net $\Phi$ over a finite directed set $I$ converges to $x$, then for each neighborhood $U$ of $x$ there is an $\alpha\in I$ such that for all $\beta\ge\alpha$ we have $\Phi_\beta\in U$. If one defines the tail $T_\alpha$ as the set of all $\Phi_\beta$ with $\beta\ge\alpha$, then this means that each $U$ around $x$ contains a tail. One also says The net $\Phi$ is eventually in $U$, since by the common upper bound in a directed set, each "line" in $I$ (which you should think of as a sequence $\gamma\le\gamma_1\le\gamma_2\le...$) eventually meets the tail $T_\alpha$.

Note that a finite directed set $I$ has a greatest element $m$ (As Asaf wrote, each finite subset of a directed set has an upper bound), but if your $(I,\le)$ is not a poset, then $m$ may not be unique. However, if $m_1$ and $m_2$ are both greatest elements, then they are equivalent ($m_1\le m_2$ and $m_2\le m_1$), and equivalent elements have always the same tails. The tail $T_m$ is then precisely the set $\{\Phi_m\mid m\text{ is a greatest element of }I\}$, call it $M$.
Now a net $\Phi$ over $I$ converges to $x$ if and only if $\Phi$ is eventually in each neighborhood of $x$, which means that each open $U$ around $x$ contains $M$.

We can summarize: A net $\Phi$ over a finite directed set $I$

• converges to $x$ if $M$ is in every neighborhood of $x$,
• has $\overline M$ as the set of cluster points.

In a $T_1$-space we can say more: The cluster points are $M$ itself, while the limit of $\Phi$ only exists if $M$ is a singleton.
Here is an example of finite net with limits and cluster points which are not limits: Let $X=\Bbb N$ with the topology generated by the sets $\{1,2,...,n\},\ n\in\Bbb N$. Let $I=(\{a,b\},\le)$, where $a<b$ and $b<a$ and $\Phi_a=2, \Phi_b=5$. Now each point $\ge5$ is a limit while each number except $1$ is a cluster point of $\Phi$. any net $\Phi$ converges to any number larger than the maximal value of $\Phi$.