Timeline for Why is a topology made up of 'open' sets?
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Jan 10, 2013 at 15:28 | comment | added | user2529 | In fact, if binary intersections exist, then all finite interesections of nonempty collections exists. But not neccessarily the whole space, i.e. the intersection of the empty collection. | |
| Jun 27, 2011 at 14:15 | history | made wiki | Post Made Community Wiki | ||
| Mar 24, 2010 at 22:21 | comment | added | M.G. | As always, the binary version of the statement does not imply in general the infinite version of the statement. In fact, since intersection is kind of refinement, it would be rather contra-intuitive to assume that the statement should be valid for infinite inttersections. But then again, of course, all this is pretty vague, and what might be intuitive for some, might look less intuitive or contra-intuitive or even wrong to others. | |
| Mar 24, 2010 at 22:16 | comment | added | M.G. | I have intentionally left out the postulate that the empty set and the whole set are also defined to be open. It seems to me rather a technical axiom and maybe there is a way to do most of the point-set-topology without it... For arbitrary unions, it seems intuitive to me that expanding an "endless set" with other "endless things" should be "endless" too (the key point is expanding). As for intersections, (my) intuition reaches only so far as to say that if A and B are "endless", then $A\cap B$ is "endless", in other words "endless" is a "shareable" property. | |
| Mar 24, 2010 at 21:40 | comment | added | LSpice | I think that this is a little dangerous—it could make it difficult for students to understand later how, say, $[a, b]$ can be open in itself. I'd also have trouble motivating why, say, arbitrary unions, but only finite intersections, of ‘endless’ sets are again ‘endless’. | |
| Mar 23, 2010 at 23:14 | history | answered | M.G. | CC BY-SA 2.5 |