Discrete subspaces of Hausdorff spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T04:42:36Z http://mathoverflow.net/feeds/question/42117 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42117/discrete-subspaces-of-hausdorff-spaces Discrete subspaces of Hausdorff spaces Pedro Perez 2010-10-14T05:48:19Z 2010-10-16T13:01:14Z <p>does every infinite hausdorff space contains a countable infinite discrete subspace?</p> http://mathoverflow.net/questions/42117/discrete-subspaces-of-hausdorff-spaces/42118#42118 Answer by brian for Discrete subspaces of Hausdorff spaces brian 2010-10-14T05:54:52Z 2010-10-14T05:54:52Z <p>Yes. Lemma 1 in <a href="http://www.emis.de/journals/HOA/IJMMS/6/197.pdf" rel="nofollow">http://www.emis.de/journals/HOA/IJMMS/6/197.pdf</a></p> http://mathoverflow.net/questions/42117/discrete-subspaces-of-hausdorff-spaces/42163#42163 Answer by Nate Eldredge for Discrete subspaces of Hausdorff spaces Nate Eldredge 2010-10-14T16:00:45Z 2010-10-14T16:00:45Z <p>Not sure whether this is "research-level", but: first show that any infinite Hausdorff space has a proper infinite closed subset. Then proceed by induction.</p> http://mathoverflow.net/questions/42117/discrete-subspaces-of-hausdorff-spaces/42385#42385 Answer by Henno Brandsma for Discrete subspaces of Hausdorff spaces Henno Brandsma 2010-10-16T13:01:14Z 2010-10-16T13:01:14Z <p>In a more general light:</p> <p>folklore theorem: Every infinite topological space contains a homeomorphic copy of one (or more) of the following 5 spaces:</p> <ol> <li>$\mathbf{N}$ in the indiscrete topology (only $\mathbf{N}$ and $\emptyset$ are open).</li> <li>$\mathbf{N}$ in the co-finite topology (only $\mathbf{N}$ and all finite sets are closed).</li> <li>$\mathbf{N}$ in the upper topology (the empty set and all sets $U(k) = \{ n \in \mathbf{N} : n \ge k \}$, $k \in \mathbf{N}$, are open).</li> <li>$\mathbf{N}$ in the lower topology ($\mathbf{N}$, $\emptyset$, and all sets $L(k) = \{ n \in \mathbf{N} : n \le k \}$, $k \in \mathbf{N}$, are open).</li> <li>$\mathbf{N}$ in the discrete topology (all subsets are open).</li> </ol> <p>As each of the spaces has the property that every infinite subspace of it is homeomorphic to the whole space, this list is minimal.</p> <p>And spaces 1-4 are not Hausdorff, which implies what you need, as being Hausdorff is hereditary.</p> <p>The nicest proof of this I know uses Ramsey's theorem (off hand I do not know a reference, who does?) using a partition of the triples or pairs of X, IIRC. </p>