Timeline for What does the 3rd axiom of topologies defined by neighbourhood mean? [closed]
Current License: CC BY-SA 3.0
23 events
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| Jun 10, 2014 at 0:26 | vote | accept | fyusuf-a | ||
| Jun 10, 2014 at 0:24 | history | edited | fyusuf-a | CC BY-SA 3.0 |
Edited so that it is not an exposition of ideas
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| Jun 8, 2014 at 17:27 | history | edited | fyusuf-a | CC BY-SA 3.0 |
added 7 characters in body
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| May 20, 2014 at 11:56 | history | edited | fyusuf-a | CC BY-SA 3.0 |
Underlined the question
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| May 8, 2014 at 17:56 | comment | added | johndoe | @Florian point well taken; I cautiously wrote 'precluding most of the local-global machinery' but your example suggests this sort of principle might be implemented in pretopological spaces as well. Next week I'll try to get my hands on "General topology" by Császár as it might contain some information about this issue. | |
| May 7, 2014 at 13:36 | comment | added | fyusuf-a | @johndoe (continuation of the previous comment) : Furthermore, the relation $xRy$ iff $\exists C\subset X,(C$ is connected $)\wedge x\in C \wedge y\in C$ is an equivalence relation (we can define the pretopology induced by $C$ taking the pretopology that makes the canonical injection $i$ a continuous function and $\forall f:Z\rightarrow C$, $f$ is continuous iff $i\circ f$ is continuous, $Z$ being a set with a pretopology). | |
| May 7, 2014 at 13:28 | comment | added | fyusuf-a | @johndoe : What do you mean by local-global machinery ? I checked for connectedness. One can, without the 3rd axiom, define a connected space the following way. For a subset $A$ of $X$, $Fr(A)=\{x\in X|\forall V\in\mathcal{V}_x,V\cap A\neq\emptyset \wedge V\cap ^cA\neq\emptyset\}$. Then $X$ is connected iff $\forall A\subset C,(A\neq\emptyset\wedge A\neq C)\Rightarrow Fr(A)\neq\emptyset$. We can check that if $X$ is connected and $A\neq\emptyset$ is a subset of $X$, then, if $A$ verifies $\forall x\in A,\exists V\in\mathcal{V}_x,V\subset A$ and $^cA$ verifies the same property, then $A=X$. | |
| May 6, 2014 at 17:16 | history | rollback | fyusuf-a |
Rollback to Revision 3
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| May 6, 2014 at 17:15 | history | rollback | fyusuf-a |
Rollback to Revision 2
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| May 6, 2014 at 6:14 | comment | added | johndoe | The third axiom says that any neighbourhood of a point $x$ is also a neighbourhood of all the points "sufficiently near" to $x$; the reason to introduce it is that axioms 1 and 2 do not put any relation among filters at different points, hence precluding most of the local-global machinery so fruitful in topology. | |
| May 5, 2014 at 21:51 | comment | added | fyusuf-a | I reformulated the question. It seems it is not a duplicate to the other question. It could even be a partial answer. But I need help with the third axiom precisely to formulate a full answer. | |
| May 5, 2014 at 20:34 | review | Reopen votes | |||
| May 7, 2014 at 2:19 | |||||
| May 5, 2014 at 20:17 | history | edited | fyusuf-a | CC BY-SA 3.0 |
Reformulated question
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| May 5, 2014 at 19:44 | history | edited | fyusuf-a | CC BY-SA 3.0 |
edited title
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| May 5, 2014 at 18:13 | history | closed |
Andrés E. Caicedo Ryan Budney Neil Strickland S. Carnahan♦ |
Needs details or clarity | |
| May 5, 2014 at 13:51 | vote | accept | fyusuf-a | ||
| May 5, 2014 at 20:17 | |||||
| May 5, 2014 at 2:20 | answer | added | Todd Trimble | timeline score: 12 | |
| May 5, 2014 at 1:38 | comment | added | David White | Possible duplicate: mathoverflow.net/questions/19152/… | |
| May 5, 2014 at 0:59 | review | Close votes | |||
| May 5, 2014 at 18:13 | |||||
| May 4, 2014 at 23:58 | comment | added | Monroe Eskew | What do you mean "need"? What is your goal and what do you think we already "need"? | |
| May 4, 2014 at 23:24 | comment | added | fyusuf-a | My question is more on terms of intuition and meaning (here limits and continuity are the most important facts as far as a topology is concerned) than formal equivalence of definitions. So the fact that there is such an equivalence does not matter in that context. This is why I asked the question. | |
| May 4, 2014 at 23:20 | comment | added | Francois Ziegler | You seem to be rediscovering Bourbaki's Proposition 2 here... | |
| May 4, 2014 at 22:54 | history | asked | fyusuf-a | CC BY-SA 3.0 |