help me understand this basic analysis proof?

Hi,

The following proof from some analysis lecture notes has me
baffled.

Theorem: Let R be a subset of a completely metrizable topological
space S. Suppose there's a subset W of R such that W is dense and
G_delta. Then R is residual (ie co-meager).

"Proof" (verbatim): If there is a subset W that is a dense G_delta,
then W is a countable intersection of open sets U_n. Furthermore, if W
is dense, then each U_n is dense. It follows that W is residual, so R
is residual.


I can see how the U_n are dense (they're supersets of W). I don't see
how it follows W is residual, and even if we assume that, I don't see
how that implies R is residual. For instance, Q is the countable
intersection Q \cap Q \cap Q... of dense subsets, but is NOT residual.

Thanks for helping,
Snis

.

Relevant Pages

  • Re: Dense sets: Question
    ... infinite countable intersection of open sets. ... a dense G_delta. ... "Understanding Godel isn't about following his formal proof. ...
    (sci.math)
  • Re: Dense sets: Question
    ... infinite countable intersection of open sets. ... Is B\C dense in R? ... "Understanding Godel isn't about following his formal proof. ...
    (sci.math)
  • Re: On how many arguments the following function can be equal to 0?
    ... This implies that S is discrete. ... furthermore conclude that S is dense (i.e. there is an element of S ... A is closed and thus has a minimal element - contradiction. ...
    (sci.math)
  • Re: On how many arguments the following function can be equal to 0?
    ... This implies that S is discrete. ... furthermore conclude that S is dense (i.e. there is an element of S ... A is closed and thus has a minimal element - contradiction. ...
    (sci.math)
  • Re: dense orbit ?
    ... So for any nontrivial compact ... a set is dense in R if every interval with distinct rational ... So in particular, x in I_0 implies x in Q_0, x in I_1 implies ... Now take the intersection of all such I_n. ...
    (sci.math)