topology by closed intervals on real line - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-19T23:49:58Z http://mathoverflow.net/feeds/question/79201 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79201/topology-by-closed-intervals-on-real-line topology by closed intervals on real line gm 2011-10-26T22:21:50Z 2011-10-26T23:12:12Z <p>As a byproduct of a discussion with a friend, I tried to find the properties of the real line if you define neighbourhoods by closed intervals. However I quickly got stuck with this problem. So to the questions.</p> <p>Can one define neighbourhoods by closed intervals? I do not see why not.</p> <p>Are the intervals of the form (a,b) now considered closed?</p> <p>Can [a,a]={a} be considered a neighbourhood?</p> <p>Is it metrizable? My guess is no but I cannot see why it shouldn't be second countable.</p> http://mathoverflow.net/questions/79201/topology-by-closed-intervals-on-real-line/79203#79203 Answer by Guillaume Brunerie for topology by closed intervals on real line Guillaume Brunerie 2011-10-26T22:32:55Z 2011-10-26T22:32:55Z <p>If you allow $[a,a]$ as a neighbourhood of $a$, then the topology is discrete and there is nothing more to say (in particular it is metrizable).</p> <p>If you only allow non-trivial closed intervals, then you have a finer topology, because every $]a,b[$ is still open. But every open set for this new topology is also open for the usual topology (because there must be a non trivial closed interval around every point, so there is also a non trivial open interval), so the topology you get is in fact exactly the usual topology of $\mathbb{R}$.</p> http://mathoverflow.net/questions/79201/topology-by-closed-intervals-on-real-line/79212#79212 Answer by Ryan Reich for topology by closed intervals on real line Ryan Reich 2011-10-26T23:12:12Z 2011-10-26T23:12:12Z <p>The union of infinitely many closed intervals is not necessarily a closed interval, i.e. $\bigcup_{n = 1}^\infty [1/n, 1] = (0, 1]$, so this is not even a topology. If you do as Guillaume Brunerie and David White are implicitly assuming and <em>generate</em> a topology via closed intervals, you get the discrete topology. If you take the above computation as a hint and look at right-half-closed intervals or left-half-closed intervals, you get what Pietro Majer said, the Sorgenfrey topology.</p>