Timeline for Dense and co-dense subsets in connected $T_2$-spaces
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 15, 2017 at 0:15 | answer | added | D.S. Lipham | timeline score: 4 | |
| Jul 14, 2017 at 16:39 | vote | accept | Dominic van der Zypen | ||
| Jul 14, 2017 at 15:36 | answer | added | Will Brian | timeline score: 12 | |
| Jul 14, 2017 at 14:40 | history | edited | Dominic van der Zypen | CC BY-SA 3.0 |
added 26 characters in body
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| Jul 14, 2017 at 13:55 | comment | added | Joel David Hamkins | If there is a base for the topology of size less than the smallest open set (a common but not universal situation), then pick a point from each set in the base, and since this will not contain any open set, the complement will be dense. One can improve this to: a base for the topology of size at most the size of the smallest open set, by combining with the transfinite recursion to build the dense set and its complement. | |
| Jul 14, 2017 at 13:47 | comment | added | Joel David Hamkins | If all open sets have the same cardinality, and this is equal to or larger than the size of a base for the topology, then we can build disjoint dense sets by transfinite recursion, by promising some points in one and other points in the other, in such a way that both are dense. | |
| Jul 14, 2017 at 13:44 | comment | added | erz | Some vague, trivial thoughts: the property is equivalent to the fact that any two dense sets must intersect. This looks like a local property: consider all the open sets, that can be decomposed as the disjoint union of their dense subsets. A countable union of such sets also has this property. Perhaps this works for ANY union as well. If our space is 2nd countable and also locally connected, then the reminder of the closure of the union will contain an open connected set $Y$ with the property that $int \overline{A}\subset \overline{int A}$, if $A\subset Y$. The latter looks very suspicious. | |
| Jul 14, 2017 at 13:25 | comment | added | Joel David Hamkins | The one-point space is an example, so probably you want to insist that there is more than one point. Or perhaps replace "connected" with "no isolated points"? | |
| Jul 14, 2017 at 13:12 | history | asked | Dominic van der Zypen | CC BY-SA 3.0 |