I have a question about nets.
I consider an infinite directed set $(A, \prec_A)$ and a net $\{x_a\}_{a \in A}$ in a compact topological space $X$. Then, it admits a convergent subnet (I consider Kelley subnets here). My question is the following: is there always exists one of these subnets $\{y_b\}_{b \in B}$ such that the directed set $(B, \prec_B)$ is infinite ?
Thank you for your help.
For $y: (B, \prec_B) \to X$ to be a subnet of $x: (A, \prec_A) \to X$, when $A$ is infinite, $B$ has to be infinite too, iff $A$ satisfies the requirement that all tails $T(a_0)=\{a: a_0 \prec a\}$ are infinite: If not, such a finite tail defines a finite subnet itself, and otherwise $B$ has a cofinal image in $A$ (under the connecting map) and in a directed set a finite set can only be cofinal if it contains a maximum of the set, while we assumed all tails to be infinite.
This regardless of compactness etc. Just consider how a finite-domained net can be a subnet of one with an infinite domain. Not very often...