Characterization of Tychonoff spaces in terms of open sets - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:08:02Z http://mathoverflow.net/feeds/question/64272 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64272/characterization-of-tychonoff-spaces-in-terms-of-open-sets Characterization of Tychonoff spaces in terms of open sets mathahada 2011-05-08T08:58:19Z 2011-05-19T08:12:32Z <p>Metrizability and complete regularity are topological properties that are, in a sense, different from the Hausdorff condition because they are not defined purely in the terms of the open sets, but rather using some external object, namely $\mathbb R$. Now a space is metrizable if it has the weak topology induced by a function $d : X \times X \rightarrow \mathbb R$ satisfying the properties of a metric, but metrization theorems tell us that an equivalent condition is that $X$ is regular and has a sigma-discrete basis (and this is purely topological). Is there a similiar characterization of Tychonoff spaces that makes no reference to $\mathbb R$ whatsoever?</p> http://mathoverflow.net/questions/64272/characterization-of-tychonoff-spaces-in-terms-of-open-sets/64284#64284 Answer by Karol Szumiło for Characterization of Tychonoff spaces in terms of open sets Karol Szumiło 2011-05-08T11:17:45Z 2011-05-08T11:17:45Z <p>A $T_1$ space $X$ is completely regular if and only if it has a basis $\mathcal{B}$ such that</p> <ol> <li>For every $x \in X$ and every $U \in \mathcal{B}$ that contains $x$, there exists $V \in \mathcal{B}$ such that $x \notin V$ and $U \cup V = X$.</li> <li>For any $U, V \in \mathcal{B}$ satisfying $U \cup V = X$, there exist $U', V' \in \mathcal{B}$ such that $X \setminus V \subset U'$, $X \setminus U \subset V'$ and $U' \cap V' = \varnothing$.</li> </ol> <p>The proof can be found in</p> <p>Frink, Orrin <em>Compactifications and semi-normal spaces.</em> Amer. J. Math. 86 1964 602–607.</p> <p>The statement I've given above is actually dual to the statement of this paper. I took it directly from Engelking's <em>General topology</em>, which is the book I strongly recommend for finding references to this kind of questions.</p> http://mathoverflow.net/questions/64272/characterization-of-tychonoff-spaces-in-terms-of-open-sets/65300#65300 Answer by KP Hart for Characterization of Tychonoff spaces in terms of open sets KP Hart 2011-05-18T07:50:46Z 2011-05-19T08:12:32Z <p>Another answer was given by Aarts and De Groot in <a href="http://cms.math.ca/10.4153/CJM-1969-010-0" rel="nofollow">Complete regularity as a separation axiom</a> Canadian Journal of Mathematics 21 1969 96–105. Frink's condition sets things up for a proof a la Urysohn's Lemma; Aarts and De Groot's condition is in terms of subbases and the proof proceeds by constructing a Hausdorff compactification.</p>