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From Engelking's book on general topology (slightly rephrased):

Definition: We say that the net S': \Sigma' \to X is finer than the net $S: \Sigma \to X$ if
1. there exists a function f: \Sigma' \to \Sigma such that for any \sigma'_0 \in \Sigma' there exists a $\sigma_0 \in \Sigma$ such that for any \sigma' \in \Sigma' \sigma'_0 \leq \sigma' implies \sigma_0 \leq f(\sigma'),
2. S \circ f = S'. We call such an $f$ a refinement.

Now the most obvious definition of a net morphism is a preorder morphism of the underlying directed sets satisfying (2). My question: is there any need to consider refinements which are not morphisms?

To be more precise:

Conjecture: For any refinement $f$ between S': \Sigma' \to X and $S: \Sigma \to X$ there exists a morphism \hat f: S' \to S such that $\hat f$ is a refinement, and \hat f(\Sigma') \subset f(\Sigma').

In plain words, this means that any refinement can be refined further using a morphism from the same net.

I can't quite get a knack of how to prove this conjecture or construct a counterexample. Also it may be that I'm overcomplicating matters and refinements can be reduced to refinement morphisms in a simpler manner. Any tips?

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    [Talk page](http://en.wikipedia.org/wiki/Talk:Subnet_%28mathematics%29) of that wikipedia article discusses several possible definitions of subnet (=finer net). Schechter's book is also mentioned there.2011-11-23

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