Re: topology with dense.


"Dave L. Renfro" <renfr1dl@xxxxxxxxx> wrote in message
news:1155503454.050431.93160@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Dave L. Renfro wrote:

Suppose there exists b in X-D such that b /= f(b).
Let V, W be disjoint neighborhoods of b, f(b).
Using continuity of f, there exist neighborhoods
V', W' of b, f(b) such that { f[V'] subset of V }
and { f[W'] subset of W }. D is dense in X and
V' intersect W' is a neighborhood of b, so there
exists d in D such that d belongs to V' intersect W'.
See if you can get a contradiction by showing that
d = f(d) belongs to both V and W.

Dave L. Renfro wrote:

Actually, something's not quite right here (3'rd sentence),
but I'll leave the details for you to work out. I've given
you the basic idea, which is probably all I should be
doing anyway.

mina_world wrote:

sorry, i can't understand that V' intersect W' is
a neighborhood of b. i need one more your advice.

In what I first wrote, change D to A (your dense set was
A, and I slipped up and used D, my letter-of-choice for
dense sets), omit the definition of V' and delete f(b)
in the 3'rd sentence [i.e. change "there exist neighborhoods
V', W' of b, f(b) such that" in the 3'rd sentence to "there
exists a neighborhood W' of b such that"], and in the 4'th
sentence change (twice) "V' intersect W'" to "V intersect W'".


yes, i see. thank you very much for your advice.


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