Is intersection of a subset with a net a subnet?

Question

Say $\{x_i\}_{i \in I}$ is a net in a set $X$ and $Y\subset X$ then, is $\{x_i\}_{i \in I}\cap Y$ a subnet? intuitively this seems true but I don't know how to approach the proof.

Answer 1

Be careful in your notations: the net is a fucntion $x: I \to X$ (often denoted for short by $\{x_i\}_{i \in I}$ and I can interpret your abuse of notation as

$x^{-1}[Y] = \{i \in I: x(i) \in Y\}$, and then you probably mean $x\restriction_{x^{-1}[Y]}$ as the subnet. It is indeed a map into $Y$, but for a net we need the domain to be directed. And for a subnet we need $x^{-1}[Y]$ to be cofinal in $I$, and then the directedness will follow from that of $I$:

So assume $x^{-1}[Y]$ is cofinal i.e.

$$\forall i \in I \exists j \in I: j \ge i \land x_j=x(j) \in Y$$

If this condition is satisfied then $x^{-1}[Y]$ is directed: if $i_1, i_2 \in x^{-1}[Y]$ we can find $i_3 \in I$ with $i_3 \ge i_1$ and $i_3 \ge i_2$, and then we have $i_4 \in x^{-1}[Y]$ so that $i_4 \ge i_3$ so by transitivity we have $i_4 \ge i_1,i_2$ as required. The inclusion map from $x^{-1}[Y]$ into $I$ shows then that $x\restriction_{x^{-1}[Y]}$ is indeed a subnet of $x$ (in both the Kelly and Willard sense, so in most texts, where one of these two is used as the definition of subnet..)

So yes iff $x^{-1}[Y]$ is cofinal in $I$.

Answer 2

No. In order for something to be a subnet it has to:

1. Be a net. How exactly $\{x_i\}_{i\in I}\cap Y$ is a net? It is a set, not a map from a directed set. A natural choice would be to define $J=\{i\in I\ |\ x_i\in Y\}$ and $j\mapsto x_j$. But then we need that $J$ to be a directed set. I.e. it has to satisfy: if $x_i\in Y$ and $x_j\in Y$ then there exists $k\in I$ such that $k\geq i$, $k\geq j$ and $x_k\in Y$. With that your $\{x_i\}_{i\in I}\cap Y$ is indeed a net.
2. There is a final monotonic function $f:J\to I$, such that our net composed with $f$ is the subnet. In other words subnets arise only by modifying the directed domain via final monotonic functions.

With that we have a simple counterexample: assume that $Y$ is such that $\{x_i\}_{i\in I}\cap Y$ is a singleton, while $I$ is a directed poset with no maximal element (e.g. $I=\mathbb{N}$ and our net is actually a sequence). Then there is no final monotonic function $J\to I$ at all, and so the induced net $(x_j)_{j\in J}$ is not a subnet of $(x_i)_{i\in I}$