proof of Tychonoff’s theorem
This is a proof in of nets. Recall the following facts:
1 - A net in converges to if and only if each coordinate converges to
3 - Every net has a universal subnet.
Proof (Tychonoff’s theorem) : Let be a net in .
Using Lemma 3 we can find a subnet of .
It is easily seen that each coordinate net is a net in .
Using Lemma 4 we see that each coordinate net converges, because is compact.
Using Lemma 1 we see that the whole net converges in .
We conclude that every net in has a convergent subnet, so, by Lemma 2, must be compact.
|Title||proof of Tychonoff’s theorem|
|Date of creation||2013-03-22 17:25:24|
|Last modified on||2013-03-22 17:25:24|
|Last modified by||asteroid (17536)|