Timeline for Paracompact but not Hausdorff
Current License: CC BY-SA 4.0
8 events
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| May 18, 2020 at 14:43 | history | edited | David White | CC BY-SA 4.0 |
Fixed minor typos since it was on the front page anyway
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Apr 1, 2010 at 13:41 | vote | accept | David Carchedi | ||
| Apr 1, 2010 at 10:55 | history | edited | Spinorbundle | CC BY-SA 2.5 |
added 945 characters in body; added 44 characters in body; edited body
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| Apr 1, 2010 at 10:52 | comment | added | villemoes | Take any open cover of the "real line" for which there are exactly one open set U_1 containing one copy of 0 (call this 0_1), and one open set U_2 containing the other copy of 0 (0_2). A partition of unity subordinate to this open covering must contain a function f_1 supported in U_1, such that f_1(0_1) = 1, since 0_1 is only contained in this open set. Similarly for 0_2. It follows by continuity that for some small epsilon, we have f_1(epsilon) + f_2(epsilon) > 1, so it is not a partition of unity. | |
| Apr 1, 2010 at 8:23 | comment | added | David Carchedi | It was in fact the question to which you linked which made me wonder this. This is not a duplicate question, since Aston insisted on a Hausdorff manifold. How can I see that the example you give is paracompact, and how can I see there is no partition of unity? | |
| Apr 1, 2010 at 8:15 | history | edited | Spinorbundle | CC BY-SA 2.5 |
added 148 characters in body
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| Apr 1, 2010 at 8:02 | history | answered | Spinorbundle | CC BY-SA 2.5 |