Timeline for Which topologies admit exactly one open singleton and have more than half of the possible subsets open?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 1 at 20:30 | comment | added | Robin Saunders | You need to try to prove both results 1) and 2) from your original conditions, and also to search for counterexamples 1') or 2') by exploring their logical consequences and then using those to constrain the possible examples. | |
| Jul 1 at 20:30 | comment | added | Robin Saunders | So if your example really is the only possible one, then there are two results you need to prove from the starting conditions: 1) all sets containing $a_0$ must be open, and 2) all nonempty sets not containing $a_0$ must not be open. Similarly, if there is another example apart from yours, then one of these results must be false for it: either 1') it has a set which contains $a_0$ and isn't open, or 2') it has a set which is open and doesn't contain $a_0$. | |
| Jul 1 at 20:12 | comment | added | Robin Saunders | Ok, but you need to test that conjecture. That means both trying to prove it (looking for reasons why it might logically follow from the conditions you imposed) and trying to disprove it (looking for a counterexample). Both of these approaches will have to use the key property of your example, which is that its nonempty open sets are precisely the sets containing $a_0$. | |
| Jun 30 at 16:20 | comment | added | amir bahadory | .yes exactly. I think there is no topology with above conditions and don't contain $a_0$ . | |
| Jun 30 at 6:40 | comment | added | Robin Saunders | So if I've understood correctly, your supervisor has asked you to work on classifying all topologies on $n$ points with exactly one open single point $a_0$, and to begin by classifying those where the topology is large ($|\tau| > 2^{n-1}$)? Is that the situation? Apart from the example you've identified, what have you tried so far? Have you considered separately the open sets which do and don't contain $a_0$, as Paul and I suggested? Where does that lead you? | |
| Jun 28 at 17:18 | comment | added | amir bahadory | Motivation: we want to solve problem about classify all topologies on finite set but for special case and we want generalize this case to another cases step by step. | |
| Jun 19 at 11:13 | comment | added | Robin Saunders | When I said "motivation", I meant mathematical motivation. Why in your thesis do you want to classify topologies with those conditions? Is it simply an exercise set by your supervisor, or are there more specific reasons why you want to consider this particular problem? Have you discussed it with your supervisor, and if so what have they suggested? Have you tried considering separately the open sets which do and don't contain $a_0$, and if so where did that lead you? | |
| Jun 18 at 13:48 | comment | added | amir bahadory | Also I need classify all these topologies | |
| Jun 18 at 9:21 | comment | added | amir bahadory | This is a question from my thesis . I want to count all topologies with above conditions. | |
| S Jun 17 at 23:27 | history | answered | Robin Saunders | CC BY-SA 4.0 | |
| S Jun 17 at 23:27 | history | made wiki | Post Made Community Wiki by Robin Saunders |