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I encountered two different definitions of subnet. The first is

Let $(I, \preceq_I ), (J,\preceq_J )$ be two directed sets and $X$ be the underlying set.$\{ \eta_j \}_{j \in J}$ is a subnet of $\{ \xi_i \}_{i \in I}$, if there exists a function $\phi: J \to I$ such that (1) $\eta_j = \xi_{\phi(j)}$ for all $j \in J$. (2) For all $i \in I$ there exists $j $ such that $\{ \phi(j') \in I \mid j´\succeq_J j\} \subseteq \{ i' \in I \mid i´\succeq_I i\}$

which is quite different from the one on page 188 Munkres' topology textbook, which states as

Let $(I, \preceq_I ), (J,\preceq_J )$ be two directed sets and $X$ be the underlying set.$\{ \eta_j \}_{j \in J}$ is a subnet of $\{ \xi_i \}_{i \in I}$, if there exists a function $\phi: J \to I$ such that (i) For all $m,n \in J$, $m \preceq_J n \implies \phi(m) \preceq_I \phi(n) $. (2) $\phi[J]$ is a cofinal in $I$.

It seems to me neither of them implies the other. What's the purpose of defining subset in two distinct ways. Are they tailor-made for different problems?

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The notes in this PDF have a decent discussion. In its terminology your first definition is of a Kelley subnet and your second of a Willard subnet. There is a third definition, of what in these notes is called an AA-subnet, that is actually better in many ways than either of these:

Let $\langle I,\preceq_I\rangle$ and $\langle J,\preceq_J\rangle$ be directed sets, and let $X$ be the underlying set. The net $\eta=\langle\eta_j:j\in J\rangle$ in $X$ is a subnet of the net $\xi=\langle\xi_i:i\in I\rangle$ if for each $A\subseteq X$, if $\xi$ is eventually in $A$, then $\eta$ is eventually in $A$. (As usual, $\eta$ is eventually in $A$ iff there is a $j_0\in J$ such that $\eta_j\in A$ whenever $j_0\preceq_J j$.)

It’s easy to see that if $\eta$ is a Willard subnet of $\xi$, then it’s a Kelley subnet, and that if it’s a Kelley subnet, then it’s an AA-subnet. Neither of these implications reverses. For example, for $n\in\Bbb N$ let $x_n=2^{-n}$ and

$$y_n=\begin{cases} 2^{-(n+1)},&\text{if }n\text{ is even}\\ 2^{-(n-1)},&\text{if }n\text{ is odd}\;; \end{cases}$$

then $\langle y_n:n\in\Bbb N\rangle$ is a Kelley subnet of $\langle x_n:n\in\Bbb N\rangle$ but not a Willard subnet.

It’s a bit harder to find an example of an AA-subnet that is not a Kelley subnet. Let $\mathscr{F}$ be the set of all functions from $\Bbb N$ to $\Bbb N$, and let $\mathscr{D}=\mathscr{F}\times\Bbb N$. For $f,g\in\mathscr{F}$ let $f\le g$ iff $f(k)\le g(k)$ for all $k\in\Bbb N$. For $\langle f,m\rangle,\langle g,n\rangle\in\mathscr{D}$ let $\langle f,m\rangle\preceq\langle g,n\rangle$ iff $f\le g$ and $m\le n$. Let $X$ be any set and $\sigma:\Bbb N\to X$ any sequence, and define

$$\nu:\mathscr{D}\to X:\langle f,m\rangle\mapsto\sigma(m)\;.$$

Suppose that $\nu$ is eventually in some $A\subseteq X$, and let $\langle f,m\rangle\in\mathscr{D}$ be such that $\nu(\langle g,n\rangle)\in A$ whenever $\langle f,m\rangle\preceq\langle g,n\rangle$. Then $\sigma(n)=\nu(\langle f,n\rangle)\in A$ whenever $n\ge m$, so $\sigma$ is an AA-subnet of $\nu$. However, there is no $\varphi:\Bbb N\to\mathscr{D}$ such that for each $\langle f,m\rangle\in\mathscr{D}$ there is an $n\in\Bbb N$ such that $\langle f,m\rangle\preceq\varphi(k)$ whenever $n\le k$, so $\sigma$ is not a Kelley subnet of $\nu$. To see this, let $\langle f_k,m_k\rangle=\varphi(k)$ for each $k\in\Bbb N$. Define $f\in\mathscr{F}$ by $f(k)=f_k(k)+1$ for each $k\in\Bbb N$; then for each $k\in\Bbb N$ we have $\langle f,0\rangle\not\preceq\langle f_k,m_k\rangle=\varphi(k)$, since $f\not\le f_k$.

AA-subnets were introduced by J.F. Aarnes and P.R. Andenæs in ‘On Nets and Filters’, which has the example that I just gave and a good discussion of why their definition is preferable to the earlier definitions.

The notes to which I linked at the top point out that in one sense it doesn’t really matter which definition we use: if $\eta$ is an AA-subnet of $\xi$, there is a Willard subnet $\nu$ of $\xi$ such that $\eta$ and $\nu$ are AA-subnets of each other and therefore have identical convergence properties.

The best discussion that I’ve seen is in Section $7$ of a set of notes by Saitulaa Naranong, ‘Translating Between Nets and Filters’; it’s still available via the WayBack Machine. Note, though, that $\Psi$ and $\Phi$ have been inadvertently interchanged in the displayed line in Definition $\mathbf{10.2}$ at the top of page $11$. The one-sentence paragraph two lines down (‘In other words ...’) is correct.

Both of the definitions that you give are modelled on the usual definition of a subsequence, and I suspect that the monotonicity requirement in the definition of Willard subnet is simply to make it look even more like the definition of subsequence. The achievement of Aarnes and Andenæs was to realize that by moving away from that model they could define subnet in a way that made a number of things work out more nicely while keeping the key property: that if a net converges to something, then so should every subnet.

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