Nets describe topologies1. Nets have the following four properties (some have already been mentioned...

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Nets describe topologies1. Nets have the following four properties (some have already been mentioned in the text):a) if x ? = x for each ?. ? /\, then x??xb) if x ? = x then every subnet of (x?)converges to x.c) if every subnet of (x?)has a subnet converging to x, then (x?) converges to x,d) [diagonal principal] if x? ? x and, for each? ? /\, a net (x?u)u?M? convergesto x?,then there is a diagonal net converging to x? i.e., the net (x?u ) ?. ? /\, u?M? ordered lexicographically by /\., then by M ? , has a subnet which converges to x.
	
Document Preview: Nets describe topologies1.  Nets have the following four properties (some have already been mentioned in the text):a) if x ?  = x for each ?. ? /\, then x??xb) if x ?  = x  then every subnet  of (x?)converges to x.c) if every subnet of (x?)has a subnet converging to x, then (x?) converges to x,d)  [diagonal  principal] if x? ? x  and,  for each ? ? /\,  a net (x?u )u?M? converges to x?,then  there is a diagonal  net converging  to x? i.e., the  net (x?u ) ?. ? /\, u?M? ordered lexicographically  by /\., then by M ? , has a subnet which converges  to x.

David · Accepted Answer

a. If xλ = x for each λ ∈ Λ, Then if U is any neighborhood of x, then we
have
xλ ∈ U∀λ
This proves
xλ → x
b. If (yλ′) : λ
′ ∈ Λ′ be a subnet of (xλ) : λ ∈ Λ, this mean there is function
h : Λ′ → Λ such that
yλ′ = xh(λ′)
and
λ′1 ≤ λ′2 ⇒ h(λ′1) ≤ h(λ′2)
Also for given λ ∈ Λ there exists λ′ ∈ Λ′ such that h(λ′) ≥ λ
So now as (xλ) converges to x, given open neighborhood of x, there exists
λ0 ∈ Λ such that
xλ ∈ U ;∀λ ≥ λ0 (1)
So we have fixed λ0 given. By the definition of subnet,

Nets describe topologies 1. Nets have the following four properties (some have already been mentioned in the text): a) if x ? = x for each ?. ? /\, then x ? ?x b) if x ? = x then every subnet of (x ?...

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