Examples of $G_\delta$ sets - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:21:53Z http://mathoverflow.net/feeds/question/57345 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57345/examples-of-g-delta-sets Examples of $G_\delta$ sets coudy 2011-03-04T11:23:49Z 2011-03-04T14:53:16Z <p>Recall that a subset A of a metric space X is a $G_\delta$ subset if it can be written as a countable intersection of open sets. This notion is related to the Baire category theorem. Here are three examples of interesting sets that are $G_\delta$ sets.</p> <ul> <li> The set of continuity points of a function $f:X\rightarrow R$. <li> The set of positively recurrent points of a continuous transformation $T: X \rightarrow X$. Recall that a point is recurrent if there exists some sequence $n_i\rightarrow \infty$ such that $T^{n_i}x \rightarrow x$. <li> The set of transitive points of a continuous transformation $T: X \rightarrow X$. Recall that a point is transitive if its orbit $\{T^n(x)\}_{n\in Z}$ is dense in $X$. </ul> <p>Question : Can you provide more examples of interesting $G_\delta$ sets ?</p> http://mathoverflow.net/questions/57345/examples-of-g-delta-sets/57348#57348 Answer by Emil Jeřábek for Examples of $G_\delta$ sets Emil Jeřábek 2011-03-04T12:11:04Z 2011-03-04T12:11:04Z <p>The set of oracles <em>A</em> (as elements of the product space $2^\omega$) such that $\mathrm P^A\ne\mathrm{NP}^A$. (This also works for nonequality of other oracle complexity classes.)</p> http://mathoverflow.net/questions/57345/examples-of-g-delta-sets/57353#57353 Answer by Todd Trimble for Examples of $G_\delta$ sets Todd Trimble 2011-03-04T13:13:20Z 2011-03-04T13:13:20Z <p>The space of nowhere differentiable continuous functions $f: [0, 1] \to \mathbb{R}$, as a subspace of the space of all continuous functions under the sup-norm topology. </p> http://mathoverflow.net/questions/57345/examples-of-g-delta-sets/57361#57361 Answer by Andreas Blass for Examples of $G_\delta$ sets Andreas Blass 2011-03-04T14:34:31Z 2011-03-04T14:34:31Z <p>In the space of all subsets of $\mathbb{N}$ (identified via characteristic functions with <code>$2^{\mathbb{N}}$</code> and topologized as the product of copies of the discrete 2-point space), the set of infinite subsets of $\mathbb{N}$ is a $G_\delta$ set. So is the set of those $A\subseteq\mathbb{N}$ for which <code>$\sum_{n\in A}1/n$</code> diverges. (The same goes with other sequences in place of $1/n$.) </p> http://mathoverflow.net/questions/57345/examples-of-g-delta-sets/57366#57366 Answer by Julián Aguirre for Examples of $G_\delta$ sets Julián Aguirre 2011-03-04T14:53:16Z 2011-03-04T14:53:16Z <p>Let $f\colon\mathbb{R}\to\mathbb{R}$ be differentiable. The set of points where $f'$ vanishes is a $G_\delta$.</p>