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net

Defines: 
subnet, Moore-Smith convergence, cluster point
Synonym: 
Moore-Smith sequence
Type of Math Object: 
Definition
Major Section: 
Reference
Groups audience: 

Mathematics Subject Classification

54A20 no label found

Comments

Hi, I am new to this forum and am not in particular very good in set theory, but I could not quite comprehend the claim made in the first paragraph of the definition for "net", that xA<=aB iff A<=B. Doesnt the mapping have to be monotonous for that claim to be valid?
Regards,
Dinesh

> ... I could not quite comprehend the
> claim made in the first paragraph of the definition for
> "net", that xA<=aB iff A<=B. Doesnt the mapping have to be
> monotonous for that claim to be valid?

The statement in question is a definition. The defined order (on X) is "induced" by \gamma and the order on A. Perhaps the difficulty is that unnecessary confusion is introduced by using \leq ambiguously to denote distinct orders.

Unless you are saying more than I think you are, "monotone" works better than "monotonous".

thanks, that helped.

I think the claim is wrong. Consider x_n = (-1)^n. (Always a good example to try!) Then this sequence is a net defined on a directed set. Since 3 =< 4, the order you mention should have -1 = (-1)^3 =< (-1)^4 = 1, yet, in a similar way, taking 4 =< 5, arrive at 1 =< -1, and if this were really a partial order (let alone a directed set!) antisymmetry gives -1 = 1, a contradiction.

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